A blood bank needs 9 people to help with a blood drive. 14 people have volunteered. Find how many different groups of 9 can be formed from the 14 volunteers.

This is a combination problem. In mathematics, a combination is a selection of items without regard to the order. In this case, we are selecting 9 people out of 14, and we don't care about the order in which they are selected.

The formula for combinations is:

C(n, k) = n! / [k!(n-k)!]

where:

• n is the total number of items,
• k is the number of items to choose,
• "!" denotes factorial, which means multiplying all positive integers up to that number.

So, in this case, n = 14 (the total number of volunteers) and k = 9 (the number of people the blood bank needs).

Substituting these values into the formula, we get:

C(14, 9) = 14! / [9!(14-9)!]

Calculating the factorials:

14! = 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 (14-9)! = 5! = 5 × 4 × 3 × 2 × 1

Substituting these values back into the formula:

C(14, 9) = (14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) / [(9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) × (5 × 4 × 3 × 2 × 1)]

Simplifying, we find that C(14, 9) = 2002.

So, there are 2002 different groups of 9 that can be formed from the 14 volunteers.