From a committee of 6 people how many sub-committees of 3 can be formed, and with 7 people and with 8 people and with n people?
SUB-C. = 6/3 = 2,
= 7/3
= 8/3
= n/3.
To find the number of sub-committees that can be formed from a larger committee, we need to use the concept of combinations. A combination is an unordered selection of a specified number of items from a larger set.
The formula to calculate the number of combinations is given by:
C(n, r) = n! / (r! * (n - r)!)
where C(n, r) represents the number of combinations, n! denotes the factorial of n, and r! represents the factorial of r.
Now let's calculate the number of sub-committees that can be formed for each case:
1. From a committee of 6 people:
Using the combination formula, we have:
C(6, 3) = 6! / (3! * (6 - 3)!)
= 6! / (3! * 3!)
= (6 * 5 * 4) / (3 * 2 * 1)
= 20
Therefore, there can be 20 sub-committees of 3 people formed from a committee of 6.
2. From a committee of 7 people:
Using the combination formula again:
C(7, 3) = 7! / (3! * (7 - 3)!)
= 7! / (3! * 4!)
= (7 * 6 * 5) / (3 * 2 * 1)
= 35
Thus, there can be 35 sub-committees of 3 people formed from a committee of 7.
3. From a committee of 8 people:
Using the combination formula:
C(8, 3) = 8! / (3! * (8 - 3)!)
= 8! / (3! * 5!)
= (8 * 7 * 6) / (3 * 2 * 1)
= 56
So, there can be 56 sub-committees of 3 people formed from a committee of 8.
4. For a committee with n people:
Using the same formula as before:
C(n, 3) = n! / (3! * (n - 3)!)
In this case, the value will depend on the specific value of n. Substitute the value of n into the formula and calculate using factorials.