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Chapter 7: Problem 382

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

Short Answer

Expert verified
There are 220 different ways to form a baseball team of 9 members from a group of 12 boys, when the position played by each member is irrelevant. This was determined using the combination formula \(^{12}C_{9} = \frac{12!}{9!(12-9)!}\).

Step by step solution

01

STEP 1: Identify the Values for n and r

In a combination, n refers to the total number of items available and r refers to the number of items to be chosen. Here, we have 12 boys (n = 12) and we need to choose 9 boys for a baseball team (r = 9).
02

STEP 2: Apply Combination Formula

Substitute the values of n and r in the combination formula \(^nC_r = \frac{n!}{r!(n-r)!}\) to calculate the number of possible teams.
03

STEP 3: Substitute and Simplify

Substituting n = 12 and r = 9 in the formula, we get: \[^{12}C_{9} = \frac{12!}{9!(12-9)!} = \frac{12!}{9!3!}\]
04

STEP 4: Calculate Factorial

Now, calculate the factorial for the numbers: \[ 12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \] \[ 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \] \[ 3! = 3 \times 2 \times 1 \] Then substitute them back into the formula and simplify: \[^{12}C_{9} = \frac{12!}{9!3!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220\]
05

STEP 5: Interpreting the Result

Hence, there are 220 different ways to form a baseball team of 9 members from a group of 12 boys.

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