A blood bank needs 12 people to help with a blood drive. 17 people have volunteered. Find how many different groups of 12 can be formed from the 17 volunteers.

To solve this problem, we need to use the concept of combinations in mathematics. A combination is a selection of items without considering the order.

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.

In this case, n = 17 (the total number of volunteers) and k = 12 (the number of people the blood bank needs).

So, we can plug these values into the formula:

C(17, 12) = 17! / [12!(17-12)!]

Calculating the factorials:

17! = 17 × 16 × 15 × 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 12! = 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 5! = 5 × 4 × 3 × 2 × 1

Substituting these values back into the formula:

C(17, 12) = 17! / (12! × 5!)

After calculating, we find that there are 6188 different groups of 12 that can be formed from the 17 volunteers.