Permutations and Combinations

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Permutations and Combinations

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Permutations and Combinations AII.12 2009 Permutations and Combinations Adapted from resources at: http://teachers.henrico.k12.va.us/math/HCPSAlgebra2/12-2.html and http://www.mathsisfun.com/combinatorics/combinations-permutations.html http://calculator.maconstate.edu/permutation/index.html

What do you call this thing? It’s a combination lock, right?

Permutations & Combinations A combination is an arrangement of items in which ORDER DOES NOT MATTER. A permutation is an arrangement of items in a particular order. Notice, ORDER MATTERS!

What should we call this thing? So it’s really a permutation lock.

How many ways can we arrange the letters ABC in groups of 2? ____ ____ AB AC BA BC CA CB How could we do it if we didn’t want to write them all out? 3 ____ How many choices are there for first blank? 3 2 For the second blank? (3)(2) = 6 Is this a permutation or a combination? It’s a permutation, because order matters. (For example, BA and AB are different things.)

Permutations The exclamation point represents a factorial. To find the number of Permutations of n items chosen r at a time, you can use the formula The exclamation point represents a factorial. n! is the product of all whole numbers less than or equal to n. For example, 5! = (5)(4)(3)(2)(1) 0! is defined to be 1.

Permutations For the ABC example, we had 3 letters, and chose 2:

If you wanted to do a permutation of 10 objects in groups of 7: In the TI-83… If you wanted to do a permutation of 10 objects in groups of 7:

Permutations =24,360 Practice: A combination lock will open when the right choice of three numbers (from 1 to 30, inclusive) is selected. How many different lock combinations are possible assuming no number is repeated? =24,360

Combinations To find the number of combinations of n items chosen r at a time, you can use the formula Or, on the calculator, you’ll find the combination choice just below the permutation choice.

Combinations For example, to choose 2 letters out of 3 in ABC, we could have: A & B A & C B & C

Combinations To play a particular card game, each player is dealt five cards from a standard 52-card deck. How many different hands are possible? Practice: Why is this a combination, and not a permutation? The order doesn’t matter. (Having 4 aces and a king is the same as getting an ace, a king, then 3 aces.)

Permutation or Combination? From a club of 24 members, a President, Vice President, Secretary, Treasurer and Historian are to be elected. In how many ways can the offices be filled? permutation

Permutation or Combination? A student must answer 3 out of 5 essay questions on a test. In how many different ways can the student select the questions? combination

Permutations or Combinations? A basketball team consists of two centers, five forwards, and four guards. In how many ways can the coach select a starting line up of one center, two forwards, and two guards? combinations Center: Forwards: Guards: Thus, the number of ways to select the starting line up is (2)(10)(6) = 120.