PROBABILITY WORKSHOP with blocks Hands-on Lwr. 2 nd’ry MATHS PROBABILITY WORKSHOP with blocks By Jon Molomby

Q : What is the waitress asking ? Review of Terms : AND and OR Q : What is the waitress asking ?

The waitress : “ Tea OR Coffee ? ” Review of Terms : AND and OR The waitress : “ Tea OR Coffee ? ”

A mathematician might say : “ ‘Tea OR Coffee ?’ …

… means Tea OR Coffee OR BOTH ”

… but Tea AND Coffee looks like this.

Do you mean Tea XOR Coffee - not BOTH ? ”

The waitress : “ Yes , that’s what I mean ! Tea XOR Coffee ? ”

1. Permutations 2. Combinations 3. Random Choice We will look at PROBABILITY using 4 blocks Green Blue Red Yellow 1. Permutations 2. Combinations 3. Random Choice You should have 4 blocks of different colour download the worksheet from www.mathswithgeoboards.com

Hands-on Lwr. 2 nd’ry MATHS 1. PERMUTATIONS with

When Caleb Gattegno taught Permutations (in “Mathematics at your Fingertips” : 1961 ) he would ask students to make different carriages for a train using Cuisenaire rods

No. of different arrangements ? Permutations Green No. of “carriages” No. of different arrangements ? Simply expressed 1 (G ) 1 ( G ) 1 !

No. of different arrangements ? Permutations Green Blue No. of “carriages” No. of different arrangements ? Simply expressed 1 (G ) 1 ( G ) 1 ! 2 (G & B ) 2 ( GB, BG ) 2 !

No. of different arrangements ? Permutations Green Blue Red No. of “carriages” No. of different arrangements ? Simply expressed 1 (G ) 1 ( G ) 1 ! 2 (G & B ) 2 ( GB, BG ) 2 ! 3 (G & B & R ) 6 ( GBR, GRB, BRG, BGR, RBG, RGB ) 3 !

No. of different arrangements ? Permutations Green Blue Red Yellow No. of “carriages” No. of different arrangements ? Simply expressed 1 (G ) 1 ( G ) 1 ! 2 (G & B ) 2 ( GB, BG ) 2 ! 3 (G & B & R ) 6 ( GBR, GRB, BRG, BGR, RBG, RGB ) 3 ! 4 (G & B & R & Y ) 24 4 !

No. of different arrangements ? Permutations Green Blue Red Yellow No. of “carriages” No. of different arrangements ? Simply expressed 1 (G ) 1 ( G ) 1 ! 2 (G & B ) 2 ( GB, BG ) 2 ! 3 (G & B & R ) 6 ( GBR, GRB, BRG, BGR, RBG, RGB ) 3 ! 4 (G & B & R & Y ) 24 4 ! 5

No. of different arrangements ? Permutations Green Blue Red Yellow No. of “carriages” No. of different arrangements ? Simply expressed 1 (G ) 1 ( G ) 1 ! 2 (G & B ) 2 ( GB, BG ) 2 ! 3 (G & B & R ) 6 ( GBR, GRB, BRG, BGR, RBG, RGB ) 3 ! 4 (G & B & R & Y ) 24 4 ! 5 120 5 !

No. of different arrangements ? Permutations Green Blue Red Yellow No. of “carriages” No. of different arrangements ? Simply expressed 1 (G ) 1 ( G ) 1 ! 2 (G & B ) 2 ( GB, BG ) 2 ! 3 (G & B & R ) 6 ( GBR, GRB, BRG, BGR, RBG, RGB ) 3 ! 4 (G & B & R & Y ) 24 4 ! 5 120 5 ! 6

No. of different arrangements ? Permutations Green Blue Red Yellow No. of “carriages” No. of different arrangements ? Simply expressed 1 (G ) 1 ( G ) 1 ! 2 (G & B ) 2 ( GB, BG ) 2 ! 3 (G & B & R ) 6 ( GBR, GRB, BRG, BGR, RBG, RGB ) 3 ! 4 (G & B & R & Y ) 24 4 ! 5 120 5 ! 6 720 6 !

n number of distinct things Permutations The General Rule n number of distinct things can be arranged in n ! ways

2) Four blocks (order important) ? PERMUTATIONS with 2) Four blocks (order important) ? Hint : (remember Q.1) 

2) Four blocks (order important) ? PERMUTATIONS with 2) Four blocks (order important) ? 4 

2) Four blocks (order important) ? PERMUTATIONS with 2) Four blocks (order important) ? 4 x 3 

2) Four blocks (order important) ? PERMUTATIONS with 2) Four blocks (order important) ? 4 x 3 x 2 

2) Four blocks (order important) ? PERMUTATIONS with 2) Four blocks (order important) ? 4 x 3 x 2 x 1 

2) Four blocks (order important) ? PERMUTATIONS with 2) Four blocks (order important) ? 4 x 3 x 2 x 1 = 24 

2) Four blocks (order important) ? PERMUTATIONS with 2) Four blocks (order important) ? 4 x 3 x 2 x 1 = 24 Ans : 24 

3) Red always first (order important) ? PERMUTATIONS with 3) Red always first (order important) ? 

3) Red always first (order important) ? PERMUTATIONS with 3) Red always first (order important) ? 1 ? ? ? Red 

3) Red always first (order important) ? PERMUTATIONS with 3) Red always first (order important) ? 1 x 3 ? ? Red 

3) Red always first (order important) ? PERMUTATIONS with 3) Red always first (order important) ? 1 x 3 x 2 ? Red 

3) Red always first (order important) ? PERMUTATIONS with 3) Red always first (order important) ? 1 x 3 x 2 x 1 = 6 Red Ans : 6 

PERMUTATIONS with 4) Red not first ? Hint : (remember Q.3) 

PERMUTATIONS with 4) Red not first ?  All possibilities minus red first 

PERMUTATIONS with 4) Red not first ?  All possibilities minus red first (from Q.3) 24 - 

PERMUTATIONS with 4) Red not first ?  All possibilities minus red first (from Q.3) 24 - 6 = 

PERMUTATIONS with 4) Red not first ?  Ans : 18 All possibilities minus red first (from Q.3) 24 - 6 = 18 Ans : 18 

PERMUTATIONS with 5) Yellow first or last ? 

PERMUTATIONS with 5) Yellow first or last ?  Mutually exclusive , so Yellow first plus Yellow last 

PERMUTATIONS with 5) Yellow first or last ? 6 + 6 = 12  Ans : 12 Mutually exclusive , so Yellow first plus Yellow last 6 + 6 = 12 Ans : 12 

6) Yellow neither first nor last ? PERMUTATIONS with 6) Yellow neither first nor last ? Hint : (remember Q.5) 

6) Yellow neither first nor last ? PERMUTATIONS with 6) Yellow neither first nor last ? All possibilities minus yellow first or last(from Q.5) 

6) Yellow neither first nor last ? PERMUTATIONS with 6) Yellow neither first nor last ? All possibilities minus yellow first or last(from Q.5) 24 - 

6) Yellow neither first nor last ? PERMUTATIONS with 6) Yellow neither first nor last ? All possibilities minus yellow first or last(from Q.5) 24 - 12 

6) Yellow neither first nor last ? PERMUTATIONS with 6) Yellow neither first nor last ? All possibilities minus yellow first or last(from Q.5) 24 - 12 = 12 Ans : 12 

PERMUTATIONS with 7) Red and Blue together ? 

PERMUTATIONS with 7) Red and Blue together ? R B 

R B so 6 x 2! PERMUTATIONS with 7) Red and Blue together ? 3 x 2 x 1 = 6 BUT RB BR R B so 6 x 2! Ans : 12 

8) Red and Blue never together ? PERMUTATIONS with 8) Red and Blue never together ? Hint : (remember Q.7) 

8) Red and Blue never together ? PERMUTATIONS with 8) Red and Blue never together ? All possibilities minus red and blue together (from Q.7) 

8) Red and Blue never together ? PERMUTATIONS with 8) Red and Blue never together ? All possibilities minus red and blue together (from Q.7) 24 - 12 = 12 Ans : 12 

Hint : PERMUTATIONS with 9) Red and Blue together, Red never next to Yellow ? Hint : (remember Q.7) 

9) Red and Blue together, Red never next to Yellow ? PERMUTATIONS with 9) Red and Blue together, Red never next to Yellow ? Red and blue together minus yellow next to RB (from Q7) (BRY or YRB) 

9) Red and Blue together, Red never next to Yellow ? PERMUTATIONS with 9) Red and Blue together, Red never next to Yellow ? Red and blue together minus yellow next to RB (from Q7) (BRY or YRB) i.e. GBRY , GYRB BRYG , YRBG 

9) Red and Blue together, Red never next to Yellow ? PERMUTATIONS with 9) Red and Blue together, Red never next to Yellow ? Red and blue together minus yellow next to RB (from Q7) (BRY or YRB) i.e. GBRY , GYRB BRYG , YRBG 12 - 4 = 8 Ans : 8 

9) Red and Blue together, Red never next to Yellow ? PERMUTATIONS with 9) Red and Blue together, Red never next to Yellow ? Checking by listing all permutations : RBGY RBYG RGBY RGYB RYBG RYGB BRGY BRYG BGRY BGYR BYRG BYGR GRBY GRYB GBRY GBYR GYRB GYBR YRBG YRGB YBRG YBGR YGRB YGBR Ans : 8 

10) Red on one end AND Green the other PERMUTATIONS with 10) Red on one end AND Green the other 

10) Red on one end AND Green the other PERMUTATIONS with 10) Red on one end AND Green the other Like this : R _ _ G R _ _ G G _ _ R 

10) Red on one end AND Green the other PERMUTATIONS with 10) Red on one end AND Green the other Possibilities : R B Y G R Y B G G B Y R G Y B R Ans : 

10) Red on one end AND Green the other R B Y G R Y B G G B Y R G Y B R PERMUTATIONS with 10) Red on one end AND Green the other R B Y G R Y B G G B Y R G Y B R Ans : 4 

10) Red on one end AND Green the other PERMUTATIONS with 10) Red on one end AND Green the other Checking by listing all permutations : RBGY RBYG RGBY RGYB RYBG RYGB BRGY BRYG BGRY BGYR BYRG BYGR GRBY GRYB GBRY GBYR GYRB GYBR YRBG YRGB YBRG YBGR YGRB YGBR Ans : 4 

11) Red on one end OR Green on an end PERMUTATIONS with 11) Red on one end OR Green on an end 

11) Red on one end OR Green on an end PERMUTATIONS with 11) Red on one end OR Green on an end Means Red on one end OR Green on an end or BOTH 

11) Red on one end OR Green on an end PERMUTATIONS with 11) Red on one end OR Green on an end Means Red on one end OR Green on an end or BOTH = All possibilities minus Red/Green in the middle 

11) Red on one end OR Green on an end PERMUTATIONS with 11) Red on one end OR Green on an end Means Red on one end OR Green on an end or BOTH = All possibilities minus Red/Green in the middle i.e. _ RG _ _ GR _ _ RG _ _ GR _ 

11) Red on one end OR Green on an end PERMUTATIONS with 11) Red on one end OR Green on an end Means Red on one end OR Green on an end or BOTH = All possibilities minus Red/Green in the middle possibilities B RG Y B GR Y Y RG B Y GR B 

11) Red on one end OR Green on an end PERMUTATIONS with 11) Red on one end OR Green on an end Means Red on one end OR Green on an end or BOTH = All possibilities minus Red/Green in the middle possibilities B RG Y B GR Y Y RG B Y GR B = 24 - 4 = 20 Ans : 20 

11)Red on one end OR Green on an end or BOTH PERMUTATIONS with 11)Red on one end OR Green on an end or BOTH Checking by listing all permutations : RBGY RBYG RGBY RGYB RYBG RYGB BRGY BRYG BGRY BGYR BYRG BYGR GRBY GRYB GBRY GBYR GYRB GYBR YRBG YRGB YBRG YBGR YGRB YGBR Ans : 20 

12) Red on an end XOR Green on an end PERMUTATIONS with 12) Red on an end XOR Green on an end 

12) Red on an end XOR Green on an end PERMUTATIONS with 12) Red on an end XOR Green on an end Means Red on an end OR Green on an end minus Red on an end AND Green on an end 

12) Red on an end XOR Green on an end PERMUTATIONS with 12) Red on an end XOR Green on an end Means Red on an end OR Green on an end (Q.11) minus Red on an end AND Green on an end (Q.10) 

12) Red on an end XOR Green on an end PERMUTATIONS with 12) Red on an end XOR Green on an end Means Red on an end OR Green on an end (Q.11) minus Red on an end AND Green on an end (Q.10) 20 - 4 = 16 Ans : 16 

12) Red on an end XOR Green on an end PERMUTATIONS with 12) Red on an end XOR Green on an end Checking by listing all permutations : RBGY RBYG RGBY RGYB RYBG RYGB BRGY BRYG BGRY BGYR BYRG BYGR GRBY GRYB GBRY GBYR GYRB GYBR YRBG YRGB YBRG YBGR YGRB YGBR Ans : 16 

PERMUTATIONS with 13) 4 Blocks in a circle 

Formula for things in a circle PERMUTATIONS with 13) 4 Blocks in a circle Formula for things in a circle = ( n – 1 ) ! 

PERMUTATIONS with 13) 4 Blocks in a circle = ( n – 1 ) ! Formula for things in a circle = ( n – 1 ) ! So the number of blocks minus 1 factorial = (4 – 1 )! = 3 ! Ans : 6 

the number of blocks factorial minus 1 = (4 – 1 )! PERMUTATIONS with 13) 4 Blocks in a circle the number of blocks factorial minus 1 = (4 – 1 )! Checking by listing all permutations : Ans : 6 

Red and Blue not next to each other PERMUTATIONS with 14) 4 Blocks in a circle : Red and Blue not next to each other 

Red and Blue not next to each other PERMUTATIONS with 14) 4 Blocks in a circle : Red and Blue not next to each other All ways in a circle minus Red/Blue together 

PERMUTATIONS with 14) 4 Blocks in a circle :  Red and Blue not next to each other All ways in a circle minus Red/Blue together 3 ! - 2 ! 2 ! = 6 – 4 = 2 Ans : 2 

Red and Blue not next to each other PERMUTATIONS with 14) 4 Blocks in a circle : Red and Blue not next to each other Checking by listing all permutations : Let 1 = Red and 2 = Blue Ans : 2 

15) 2 selected from 4 blocks (order important)? PERMUTATIONS with 15) 2 selected from 4 blocks (order important)? 

15) 2 selected from 4 blocks (order important)? PERMUTATIONS with 15) 2 selected from 4 blocks (order important)? Using the formula for PERMUTATIONS : 

15) 2 selected from 4 blocks (order important)? PERMUTATIONS with 15) 2 selected from 4 blocks (order important)? Using the formula for PERMUTATIONS : 

15) 2 selected from 4 blocks (order important)? PERMUTATIONS with 15) 2 selected from 4 blocks (order important)? Using the formula for PERMUTATIONS : 4P2 = 

15) 2 selected from 4 blocks (order important)? PERMUTATIONS with 15) 2 selected from 4 blocks (order important)? Using the formula for PERMUTATIONS : 4P2 = 

15) 2 selected from 4 blocks (order important)? PERMUTATIONS with 15) 2 selected from 4 blocks (order important)? Using the formula for PERMUTATIONS : 4P2 = Ans : 12 

15) 2 selected from 4 blocks (order important)? PERMUTATIONS with 15) 2 selected from 4 blocks (order important)? An easier way ? 

15) 2 selected from 4 blocks (order important)? PERMUTATIONS with 15) 2 selected from 4 blocks (order important)? An easier way ? Yes Based on nPk = n factorial to the first k no. of factors 

15) 2 selected from 4 blocks (order important)? PERMUTATIONS with 15) 2 selected from 4 blocks (order important)? An easier way ? Based on nPk = n factorial to the first k no. of factors 4P2 = 4 factorial to the first two factors 

15) 2 selected from 4 blocks (order important)? PERMUTATIONS with 15) 2 selected from 4 blocks (order important)? An easier way ? Based on nPk = n factorial to the first k no. of factors 4P2 = 4 factorial to the first two factors 4 x 3 = 12 

15) 2 selected from 4 blocks (order important)? PERMUTATIONS with 15) 2 selected from 4 blocks (order important)? An easier way ? Based on nPk = n factorial to the first k no. of factors 4P2 = 4 factorial to the first two factors 4 x 3 = 12 Ans : 12 

16) 3 selected from 4 blocks (order important)? PERMUTATIONS with 16) 3 selected from 4 blocks (order important)? 

16) 3 selected from 4 blocks (order important)? PERMUTATIONS with 16) 3 selected from 4 blocks (order important)? Using the formula for PERMUTATIONS : 

16) 3 selected from 4 blocks (order important)? PERMUTATIONS with 16) 3 selected from 4 blocks (order important)? Using the formula for PERMUTATIONS : 

16) 3 selected from 4 blocks (order important)? PERMUTATIONS with 16) 3 selected from 4 blocks (order important)? Using the formula for PERMUTATIONS : 4P3 = 

16) 3 selected from 4 blocks (order important)? PERMUTATIONS with 16) 3 selected from 4 blocks (order important)? Using the formula for PERMUTATIONS : 4P3 = 

16) 3 selected from 4 blocks (order important)? PERMUTATIONS with 16) 3 selected from 4 blocks (order important)? Using the formula for PERMUTATIONS : 4P3 = Ans : 24 

16) 3 selected from 4 blocks (order important)? PERMUTATIONS with 16) 3 selected from 4 blocks (order important)? An easier way ? 

16) 3 selected from 4 blocks (order important)? PERMUTATIONS with 16) 3 selected from 4 blocks (order important)? An easier way ? Yes Based on nPk = n factorial to the first k no. of factors 

16) 3 selected from 4 blocks (order important)? PERMUTATIONS with 16) 3 selected from 4 blocks (order important)? An easier way ? Based on nPk = n factorial to the first k no. of factors 4P3 = 4 factorial to the first three factors 

16) 3 selected from 4 blocks (order important)? PERMUTATIONS with 16) 3 selected from 4 blocks (order important)? An easier way ? Based on nPk = n factorial to the first k no. of factors 4P3 = 4 factorial to the first three factors 4 x 3 x 2 = 

16) 3 selected from 4 blocks (order important)? PERMUTATIONS with 16) 3 selected from 4 blocks (order important)? An easier way ? Based on nPk = n factorial to the first k no. of factors 4P3 = 4 factorial to the first three factors 4 x 3 x 2 = 24 Ans : 24 

Hands-on Lwr. 2 nd’ry MATHS 2. COMBINATIONS with

17) 2 chosen from 4 blocks (order not important)? COMBINATIONS with 17) 2 chosen from 4 blocks (order not important)? 

17) 2 chosen from 4 blocks (order not important)? COMBINATIONS with 17) 2 chosen from 4 blocks (order not important)? Using the formula for Combinations : 

17) 2 chosen from 4 blocks (order not important)? COMBINATIONS with 17) 2 chosen from 4 blocks (order not important)? Using the formula for Combinations : 

17) 2 chosen from 4 blocks (order not important)? COMBINATIONS with 17) 2 chosen from 4 blocks (order not important)? Using the formula for Combinations : 

17) 2 chosen from 4 blocks (order not important)? COMBINATIONS with 17) 2 chosen from 4 blocks (order not important)? Using the formula for Combinations : Ans : 6 

17) 2 chosen from 4 blocks (order not important)? COMBINATIONS with 17) 2 chosen from 4 blocks (order not important)? An easier way ? 

17) 2 chosen from 4 blocks (order not important)? COMBINATIONS with 17) 2 chosen from 4 blocks (order not important)? An easier way ? Yes … Using Pascal’s Triangle 

17) 2 chosen from 4 blocks (order not important)? COMBINATIONS with 17) 2 chosen from 4 blocks (order not important)? An easier way ? Using Pascal’s Triangle 

17) 2 chosen from 4 blocks (order not important)? COMBINATIONS with 17) 2 chosen from 4 blocks (order not important)? An easier way ? Using Pascal’s Triangle 

17) 2 chosen from 4 blocks (order not important)? COMBINATIONS with 17) 2 chosen from 4 blocks (order not important)? An easier way ? 4C2 Using Pascal’s Triangle 

17) 2 chosen from 4 blocks (order not important)? COMBINATIONS with 17) 2 chosen from 4 blocks (order not important)? An easier way ? 4C2 Using Pascal’s Triangle 

17) 2 chosen from 4 blocks (order not important)? COMBINATIONS with 17) 2 chosen from 4 blocks (order not important)? An easier way ? 4C2 Using Pascal’s Triangle 

17) 2 chosen from 4 blocks (order not important)? COMBINATIONS with 17) 2 chosen from 4 blocks (order not important)? An easier way ? 4C2 Using Pascal’s Triangle Ans : 6 

18) 3 chosen from 4 blocks (order not important)? COMBINATIONS with 18) 3 chosen from 4 blocks (order not important)? An easier way ? Using Pascal’s Triangle 

18) 3 chosen from 4 blocks (order not important)? COMBINATIONS with 18) 3 chosen from 4 blocks (order not important)? An easier way ? 4C3 Using Pascal’s Triangle 

18) 3 chosen from 4 blocks (order not important)? COMBINATIONS with 18) 3 chosen from 4 blocks (order not important)? An easier way ? 4C3 Using Pascal’s Triangle 

18) 3 chosen from 4 blocks (order not important)? COMBINATIONS with 18) 3 chosen from 4 blocks (order not important)? An easier way ? 4C3 Using Pascal’s Triangle 

18) 3 chosen from 4 blocks (order not important)? COMBINATIONS with 18) 3 chosen from 4 blocks (order not important)? An easier way ? 4C3 Using Pascal’s Triangle Ans : 4 

19) 4 chosen from 4 blocks (order not important)? COMBINATIONS with 19) 4 chosen from 4 blocks (order not important)? An easier way ? Using Pascal’s Triangle 

19) 4 chosen from 4 blocks (order not important)? COMBINATIONS with 19) 4 chosen from 4 blocks (order not important)? An easier way ? 4C4 Using Pascal’s Triangle 

19) 4 chosen from 4 blocks (order not important)? COMBINATIONS with 19) 4 chosen from 4 blocks (order not important)? An easier way ? 4C4 Using Pascal’s Triangle 

19) 4 chosen from 4 blocks (order not important)? COMBINATIONS with 19) 4 chosen from 4 blocks (order not important)? An easier way ? 4C4 Using Pascal’s Triangle Ans : 1 

20) 0 chosen from 4 blocks (order not important)? COMBINATIONS with 20) 0 chosen from 4 blocks (order not important)? An easier way ? Using Pascal’s Triangle 

20) 0 chosen from 4 blocks (order not important)? COMBINATIONS with 20) 0 chosen from 4 blocks (order not important)? An easier way ? 4C0 Using Pascal’s Triangle 

20) 0 chosen from 4 blocks (order not important)? COMBINATIONS with 20) 0 chosen from 4 blocks (order not important)? An easier way ? 4C0 Using Pascal’s Triangle 

20) 0 chosen from 4 blocks (order not important)? COMBINATIONS with 20) 0 chosen from 4 blocks (order not important)? An easier way ? 4C0 Using Pascal’s Triangle Ans : 1 

21) 1 chosen from 4 blocks (order not important)? COMBINATIONS with 21) 1 chosen from 4 blocks (order not important)? An easier way ? Using Pascal’s Triangle 

21) 1 chosen from 4 blocks (order not important)? COMBINATIONS with 21) 1 chosen from 4 blocks (order not important)? An easier way ? 4C1 Using Pascal’s Triangle 

21) 1 chosen from 4 blocks (order not important)? COMBINATIONS with 21) 1 chosen from 4 blocks (order not important)? An easier way ? 4C1 Using Pascal’s Triangle 

21) 1 chosen from 4 blocks (order not important)? COMBINATIONS with 21) 1 chosen from 4 blocks (order not important)? An easier way ? 4C1 Using Pascal’s Triangle Ans : 4 

Another handy trick with Combinations COMBINATIONS with Another handy trick with Combinations nCr = nCn-r e.g. 20C18 = 20C2 

Hands-on Lwr. 2 nd’ry MATHS 3. Random Choice without replacement

RANDOM CHOICE with Put the 4 blocks inside your hands :  What are the chances if taken at random …. ? 

RANDOM CHOICE with Put the 4 blocks inside your hands : 22)  Red (taking 1) 

RANDOM CHOICE with Ans : 1 out of 4 Put the 4 blocks inside your hands : 22) Red (taking 1) Ans : 1 out of 4 

RANDOM CHOICE with Put the 4 blocks inside your hands : 23)  Red or Blue (taking 1) 

RANDOM CHOICE with Ans : 2 out of 4 = 1 out of 2 Put the 4 blocks inside your hands : 23) Red or Blue (taking 1) Ans : 2 out of 4 = 1 out of 2 

RANDOM CHOICE with Put the 4 blocks inside your hands : 24)  Red then Blue (taking 2 without replacement) 

RANDOM CHOICE with Ans : 𝟏 𝟒 × 𝟏 𝟑 = 𝟏 𝟏𝟐 = 1 out of 12 Put the 4 blocks inside your hands : 24) Red then Blue (taking 2 without replacement) Ans : 𝟏 𝟒 × 𝟏 𝟑 = 𝟏 𝟏𝟐 = 1 out of 12 

RANDOM CHOICE with Put 8 blocks inside your hands : 25)  Red (taking 1) 

RANDOM CHOICE with Ans : 2 out of 8 = 1 out of 4 Put 8 blocks inside your hands : 25) Red (taking 1) Ans : 2 out of 8 = 1 out of 4 

RANDOM CHOICE with Put 8 blocks inside your hands : 26)  Blue or Green (taking 1) 

RANDOM CHOICE with Ans : 4 out of 8 = 1 out of 2 Put 8 blocks inside your hands : 26) Blue or Green (taking 1) Ans : 4 out of 8 = 1 out of 2 

RANDOM CHOICE with Put 8 blocks inside your hands : 27)  Blue, Blue (taking 2 without replacement) 

RANDOM CHOICE with Ans : 𝟏 𝟒 × 𝟏 𝟕 = 𝟏 𝟐𝟖 1 out of 28 Put 8 blocks inside your hands : 27) Blue, Blue (taking 2 without replacement) Ans : 𝟏 𝟒 × 𝟏 𝟕 = 𝟏 𝟐𝟖 1 out of 28 

RANDOM CHOICE with Put 8 blocks inside your hands : 28)  Any 2 of the same colour (taking 2 without replacement) 

RANDOM CHOICE with Ans : 𝟏× 𝟏 𝟕 = 𝟏 𝟕 1 out of 7 Put 8 blocks inside your hands : 28) Any 2 of the same colour (taking 2 without replacement) Ans : 𝟏× 𝟏 𝟕 = 𝟏 𝟕 1 out of 7 

RANDOM CHOICE with Put 8 blocks inside your hands : 29)  Any 2 of different colour (taking 2 without replacement) 

RANDOM CHOICE with Ans : 𝟏× 𝟔 𝟕 = 𝟔 𝟕 6 out of 7 Put 8 blocks inside your hands : 29) Any 2 of different colour (taking 2 without replacement) Ans : 𝟏× 𝟔 𝟕 = 𝟔 𝟕 6 out of 7 

Permutation question : How many ways can you order PERMUTATIONS with 30) Permutation question : How many ways can you order 8 blocks if there are 2 red, 2 blue, 2 yellow, 2 green ? 

Permutation question : How many ways can you order PERMUTATIONS with 30) Permutation question : How many ways can you order 8 blocks if there are 2 red, 2 blue, 2 yellow, 2 green ? Ans : 8! / (2! 2! 2! 2!) = 2520 ways 

PERMUTATIONS with 31) 8 blocks in a circle if there are  Permutation question : How many ways can you order 8 blocks in a circle if there are 2 red, 2 blue, 2 yellow, 2 green ? 

PERMUTATIONS with 31) Ans : 7! / (2! 2! 2! 2!) = 315 ways Permutation question : How many ways can you order 8 blocks in a circle if there are 2 red, 2 blue, 2 yellow, 2 green ? Ans : 7! / (2! 2! 2! 2!) = 315 ways 

PERMUTATIONS with 32) 2 blocks (order is important) from 5 blocks 32)   Permutation question : How many ways can you select 2 blocks (order is important) from 5 blocks (green, green, blue, red, yellow) [note : 5P2 over 2 ! is not correct]

Ans : 4P2 + 1 = 13 (see next slide) PERMUTATIONS with 32)   Permutation question : How many ways can you select 2 blocks (order is important) from 5 blocks (green, green, blue, red, yellow) [note : 5P2 over 2 ! is not correct] Ans : 4P2 + 1 = 13 (see next slide)

PERMUTATIONS with 32)  There is no difference between Green 1 and Green 2 So the permutations are : GB, GR, GY, BR, BG, BY, RG, RY, RB, YG, YB, YR. i.e. 4P2 = 12 + GG 1 Ans : 13

33) Some groups have 4 blocks, COMBINATIONS 33)   Some groups have 4 blocks, and some groups have 4 triangular prisms. In how many ways can I choose one block and one triangular prism ?

33) Some groups have 4 blocks, COMBINATIONS with 33)   Some groups have 4 blocks, and some groups have 4 triangular prisms. In how many ways can I choose one block and one triangular prism ? Ans : 16 ways (see next slide)

33) Based on the Fundamental Counting Principle COMBINATIONS 33)   Based on the Fundamental Counting Principle If I have m kind of things and n kind of things, I can combine one of each in m x n ways 4 blocks x 4 triangular prisms = 16 ways Ans : 16 ways

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