7.3 Combinations Math 30-11

A combination is a selection of a group of objects taken from a larger pool for which the kinds of objects selected is of importance but not the order in which they were selected. How many arrangements are there for the letters ABC? ABC ACB BCA BAC CAB CBA When the order of the letters is important there are six distinct arrangements or permutations. However, if order is not important and all you wanted was a grouping of ABC, there is only one way, or one combination. The number of combinations of n items taken r at a time is: Combinations When order matters, you have permutations. When order does not matter, you have combinations. n r Math 30-12

1. Evaluate the following. = A committee of four students is to be chosen from a group of 10 students. In how many ways can this be done? 10 C 4 = 210 The committee of four can be selected in 210 ways. Finding the Number of Combinations Math 30-13

3. a) A company is hiring people to fill five identical positions. There are 12 applicants. In how many ways can the company fill the five positions? 12 C 5 = 792 (The number of combinations of 12 taken five at a time is 792). b) The company wants to hire applicant A and four of the other applicants. How many ways can the five positions be filled? 1 C 1 x 11 C 4 = 330 With the selection of applicant A and four others, there are 330 ways of filling the positions. Finding the Number of Combinations The company can fill the five positions 792 ways. 4. A math class has 18 male students and 19 female students. A committee of four male and three female is to be selected. How many ways can this be done? 18 C 4 x 19 C 3 = Math 30-14

4. a) There are seven books on a shelf. In how many ways can five or more books be selected? Select 5 or 6 or 7: 7 C C C 7 = = 29 There are 29 ways to select five or more books. b) If zero to seven books were to be selected, how many ways could this be done? 7C0 7C0 + 7C1 7C1 + 7 C 2 + 7C3 7C3 + 7C4 7C4 + 7 C 5 + 7C6 7C6 + 7 C 7 = 128 To find the number of ways to select from zero to n objects, use 2 n. 5. How many ways can one or more of five different toys be selected? 2 n - 5 C 0 = = = 31 There are 31 ways to select one and five of five toys. 2 7 = 128 Connecting words and, or, at least, at most, and no more than. Alternative Strategy: Math 30-15

6. There are seven women and five men applying for four positions with a company. The hiring committee wants to hire at least one woman. How many different ways can the four positions be filled? 1 woman and 3 men ( 7 C 1 x 5 C 3 ) = 490 The four positions can be filled 490 different ways. Alternative Strategy: Take the total number of combinations and subtract the combinations containing no women. 12 C C 4 x 7 C 0 = 490 Finding the Number of Combinations or 2 women and 2 men or 3 women and 1 man or 4 women and zero men + ( 7 C 2 x 5 C 2 )+ ( 7 C 3 x 5 C 1 )+ ( 7 C 4 x 5 C 0 ) Math 30-16

Finding the Number of Combinations 7. A math class has 18 male students and 19 female students. A committee of seven is to be selected. How many ways can this be done, if there must be at least one female on the committee? 37 C C 7 x 19 C 0 = A committee of six is to be chosen from three girls and seven boys. Two particular boys must be on the committee. Find the number of ways of selecting the committee. 2 C 2 x 8 C 4 = How many five card hands can be dealt from a standard deck of 52 cards if: a) each hand must contain two aces? b) each hand must contain three red cards? 4 C 2 x 48 C 3 = C 3 x 26 C 2 = Math 30-17

7. There are eight points in a coordinate plane, no three points are collinear. a) How many line segments can be drawn? 8C28C2 = 28 b) How many triangles can be drawn? 8C38C3 = 56 c) How many quadrilaterals can be drawn? 8C48C4 = 70 Problem Solving Math 30-18

8. How many diagonals are there in a hexagon? in an octagon? Hexagon: 6 C = 9 Octagon: 8 C = 20 in an n-sided polygon? n C 2 - n Finding the Number of Combinations Math 30-19

1. A basketball league has eight teams. Each team must play each other team four times during the season. How many games must be scheduled? 8 C 2 x 4 = games must be scheduled. 2. Solve the equation n C 2 = 10 for n. n C 2 = 10 n 2 - n = 20 n 2 - n - 20 = 0 (n - 5)(n + 4) = 0 n = 5 or n = -4 Therefore, n = 5. Diploma Questions Math

3. Show that 10 C 4 = 10 C 6 10 C 4 = 10 C = 210 Therefore 10 C 4 = 10 C 6. Therefore n C r = n C (n - r). Solving Combinations n C 5 = n C 7 n = C 5 = 14 C r r = 9 Solve: Math

Solving Problems with Combinations 4. If there are 190 handshakes in a room, and each person shook every other person’s hand one time, how many people are in the room? Therefore there would be 20 people in the room. Math

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