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Chapter 2: Problem 82

How many baseball teams of nine members can be chosen from among I twelve boys, without regard to the position played by each member?

Short Answer

Expert verified
220 different baseball teams of nine members can be chosen from among twelve boys.

Step by step solution

01

Set up the combination formula

Write down the combination formula and plug in the values of n and k. \(C(n, k) = \frac{n!}{k!(n-k)!}\) Here, n = 12 and k = 9. Let's substitute these values into the formula: \(C(12, 9) = \frac{12!}{9!(12-9)!}\)
02

Calculate factorials

Calculate the factorial of 12!, 9!, and (12-9)! in the formula. \[12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\] \[9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\] \[(12-9)! = 3! = 3 \times 2 \times 1\]
03

Simplify the equation

Use the calculated factorial values to simplify the formula: \[C(12, 9) = \frac{12 \times 11 \times 10 \times 9!}{9! \times (3 \times 2 \times 1)}\] Notice that the 9! from numerator and denominator will cancel each other out: \[C(12, 9) = \frac{12 \times 11 \times 10}{3 \times 2 \times 1}\]
04

Calculate the final answer

Perform the calculations to find out the number of baseball teams possible: \[C(12, 9) = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = \frac{1320}{6} = 220\] So, 220 different baseball teams of nine members can be chosen from among twelve boys.

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