Question

There are 20 students on the school's student council. A special homecoming dance committee is to be formed by randomly selecting 7 students from student council. How many possible committees can be formed?

Answer 1

Using the combinations formula, there can be 77,520 different committees formed by selecting 7 students from a student council of 20 students.

Step-by-step explanation:

To determine the number of possible committees that can be formed by selecting 7 students from a group of 20 students, we will use the combinations formula since the order of selection does not matter. This is a classic example of a combinatorial problem where we are choosing a subgroup from a larger group without regard to the order in which they are chosen.

The formula for combinations is as follows:

C(n, k) = n! / (k! * (n - k)!)

Where:

n is the total number of items,

k is the number of items to choose,

! indicates factorial, which means the product of all positive integers up to that number. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.

Applying this formula to our problem:

C(20, 7) = 20! / (7! * (20 - 7)!) = 20! / (7! * 13!) = (20 x 19 x 18 x 17 x 16 x 15 x 14) / (7 x 6 x 5 x 4 x 3 x 2 x 1)

After simplifying the factorial expressions and canceling out common factors, we find the number of possible committees that can be formed.

Therefore, there are 77520 possible committees that can be formed from a student council of 20 students by selecting 7.

Answer 2

Total 776,520 committees are possible.

Step-by-step explanation:

Since, The total number of students = 20,

The number of students in the committee = 7,

Hence, the possible number of committees can be formed = Total combination of 7 students out of 20 students

`=^(20)C_7`

`=(20!)/(7!* (20-7)!)`

`=(20!)/(5040* 13!)`

`=(20* 19* 18* 17* 16* 15* 14)/(5040)`

`=(390700800)/(5040)=776520`

⇒ 776,520 committees are possible.