Ms. Bell's mathematics class consists of 6 sophomores, 11 juniors, and 13 seniors. How many different ways can Ms. Bell create a 5-member committee of seniors if each senior has an equal chance of being selected?
Question
Ms. Bell's mathematics class consists of 6 sophomores, 11 juniors, and 13 seniors. How many different ways can Ms. Bell create a 5-member committee of seniors if each senior has an equal chance of being selected?
Solution
To solve this problem, we need to use the concept of combinations in mathematics. A combination is a selection of items without regard to the order in which they are selected.
The formula for combinations is:
C(n, r) = n! / [r!(n-r)!]
where:
- n is the total number of items,
- r is the number of items to choose,
- "!" denotes factorial, which means multiplying all positive integers up to that number.
In this case, n is the number of seniors, which is 13, and r is the size of the committee, which is 5.
So, the number of ways Ms. Bell can create a 5-member committee of seniors is:
C(13, 5) = 13! / [5!(13-5)!]
Calculating the factorials:
13! = 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 5! = 5 × 4 × 3 × 2 × 1 (13-5)! = 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
Substituting these values into the formula:
C(13, 5) = 13! / [5!(13-5)!] = (13 × 12 × 11 × 10 × 9) / (5 × 4 × 3 × 2 × 1) = 1287
So, there are 1287 different ways Ms. Bell can create a 5-member committee of seniors.
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