Did you know?
Did you know that the number of different groups that can be formed from a larger group can be calculated using a mathematical concept called combinations? In this scenario, a blood bank needs 9 people for a blood drive, and luckily, 14 people have volunteered. To find out how many different groups of 9 can be formed from the 14 volunteers, we can use the combination formula.
Step 1: Identify the total number of volunteers (14) and the number of people needed per group (9).
Step 2: Plug these values into the combination formula: nCr = n! / (r!(n-r)!), where n is the total number of items (14), r is the number of items chosen (9), and ! represents a factorial (the product of an integer and all positive integers less than it).
Step 3: Calculate the factorials for the given values: 14! / (9!(14-9)!). Simplifying further, we have 14! / (9! * 5!).
Step 4: Evaluate the factorials. 14! = 14 * 13 * 12 * 11 * 10 * 9! and 5! = 5 * 4 * 3 * 2 * 1.
Step 5: Substitute the factorials back into the combination formula: (14 * 13 * 12 * 11 * 10 * 9!) / (9! * (5 * 4 * 3 * 2 * 1)).
Step 6: Notice that the 9! terms cancel out in the numerator and denominator, as well as many other factors: (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)).
Step 7: Simplify further: (14 * 13 * 12 * 11 * 10) / (5 * 4 * 3 * 2 * 1).
Step 8: Calculate the products in the numerator and denominator: 240240 / 120.
Step 9: Perform the division: 2002.
Therefore, from the 14 volunteers, there are exactly 2002 different groups of 9 people that can be formed for the blood drive.
