In a given city there are 28 reporters of which 16 are men. If the committee of 4 reporters is to be formed
A. How many different committees can be formed?
B. How many different committees contain 3 men?
C. How many different commitrees contain at least 2 men?
D. How many different committees contain at most 3 men?
To answer these questions, we can use combinatorics principles. Let's break down each question step by step:
A. How many different committees can be formed?
To calculate the number of different committees, we need to calculate the combinations of 28 reporters taken 4 at a time (without considering gender). We can use the formula for combinations:
nCr = n! / ((n-r)! * r!)
In this case, n = 28 (the total number of reporters) and r = 4 (the number of reporters in each committee). Plugging in these values, the calculation becomes:
28C4 = 28! / ((28-4)! * 4!) = 28! / (24! * 4!)
Calculating this would give you the answer to question A.
B. How many different committees contain 3 men?
To calculate the number of committees with 3 men, we need to consider the number of combinations of 16 men taken 3 at a time (while the fourth reporter is chosen from the remaining group of 12 women). Using the same formula with n = 16 and r = 3, the calculation becomes:
16C3 = 16! / ((16-3)! * 3!)
Calculating this would give you the answer to question B.
C. How many different committees contain at least 2 men?
To calculate the number of committees with at least 2 men, we need to consider two scenarios: committees with exactly 2 men and committees with 3 men.
To calculate the number of committees with exactly 2 men, we use the combinations of 16 men taken 2 at a time (while the other two reporters are chosen from the remaining group of 12 women). Using the same formula with n = 16 and r = 2, the calculation becomes:
16C2 = 16! / ((16-2)! * 2!)
To calculate the number of committees with 3 men, we use the calculation from question B.
To account for both scenarios, you add the number of committees with exactly 2 men and the number of committees with 3 men.
D. How many different committees contain at most 3 men?
To calculate the number of committees with at most 3 men, you need to consider committees with 0 men, 1 man, 2 men, and 3 men.
To calculate the number of committees with 0 men, you use the combinations of 12 women taken 4 at a time. Using the same formula with n = 12 and r = 4, the calculation becomes:
12C4 = 12! / ((12-4)! * 4!)
To calculate the number of committees with 1 man, you multiply the combinations of 16 men taken 1 at a time (while the other three reporters are chosen from the remaining group of 12 women) by the total number of combinations of those three remaining reporters. You can represent this mathematically as:
(16C1) * (12C3)
To calculate the number of committees with 2 men, you use the calculation from question C.
To calculate the number of committees with 3 men, you use the calculation from question B.
To account for all four scenarios, you sum up the number of committees with 0 men, 1 man, 2 men, and 3 men.
By following these steps, you can find the answers to questions B, C, and D using combinatorics principles.