Question

A school committee consists of 2 teachers and 4 students. The number of different committees that can be formed from 5 teachers and 10 students is
A. 10
B. 15
C. 2100
D. 8

Answer 1

Hint: Here in this question we have to make the group which consists of 2 teachers and 4 students from 5 teachers and 10 students. To find the number of committees we will apply the combination formula.

Complete step-by-step answer:
Now we have to form the groups from the given data. Here we use the topic combination to make the committees of 2 teachers and 4 students from 5 teachers and 10students. Combination means grouping of items. It is defined as \[C(n,r) = {}^n{C_r} = \dfrac{{n!}}{{(n - r)!r!}}\] where \[n\] is number of items and \[r\] is number of items selected from the given items
Therefore, now we have to select the 2 teachers and 4 students from 5 teachers and 10 students and we use the above formula. The number of possibilities to select 2 teachers from the 5 teachers is \[C(5,2) = {}^5{C_2} = \dfrac{{5!}}{{(5 - 2)!2!}}\]
 \[ \Rightarrow C(5,2) = {}^5{C_2} = \dfrac{{5!}}{{3!2!}}\]
 \[ \Rightarrow C(5,2) = \dfrac{{5 \times 4 \times 3 \times 2 \times 1}}{{(3 \times 2 \times 1)(2 \times 1)}}\]
Since we know that \[n! = n \times (n - 1) \times (n - 2) \times ... \times 1\] . Now we cancel the similar terms which are in both denominator and the numerator. Therefore, we have
 \[ \Rightarrow C(5,2) = \dfrac{{5 \times 4}}{{2 \times 1}}\]
 \[ \Rightarrow C(5,2) = \dfrac{{20}}{2}\]
 \[ \Rightarrow C(5,2) = 10\]
Therefore, we have 10 ways of possibilities to select 2 teachers from 5 teachers
Now we have to select 4 students from 10 students. The number of ways of possibilities to select 4 students from 10 students is \[C(10,4) = {}^{10}{C_4} = \dfrac{{10!}}{{(10 - 4)!4!}}\]
 \[ \Rightarrow C(10,4) = {}^{10}{C_4} = \dfrac{{10!}}{{6!4!}}\]
 \[ \Rightarrow C(10,4) = \dfrac{{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{(6 \times 5 \times 4 \times 3 \times 2 \times 1)(4 \times 3 \times 2 \times 1)}}\]
Since we know that \[n! = n \times (n - 1) \times (n - 2) \times ... \times 1\] . Now we cancel the similar terms which are in both denominator and the numerator. Therefore, we have
 \[ \Rightarrow C(10,4) = \dfrac{{10 \times 9 \times 8 \times 7}}{{4 \times 3 \times 2 \times 1}}\]
 \[ \Rightarrow C(10,4) = \dfrac{{5040}}{{24}}\]
 \[ \Rightarrow C(10,4) = 210\]
Therefore, we have 210 ways of possibilities to select 4 students from 10 students.
Now we have to find the school committee consisting of 2 teachers and 4 students together from 5 teachers and 10 students. We use multiplication principle of counting, it is defined as \[{n_1}*{n_2}\] where \[{n_1}\] and \[{n_2}\] are possible outcomes. Here \[{n_1}\] is \[C(5,2)\] and \[{n_2}\] is \[C(10,4)\]
Therefore, we have \[C(5,2)*C(10,4)\]
 \[ \Rightarrow 10*210\]
 \[ \Rightarrow 2100\]
Therefore 2100 ways of possibilities we have to form a school committee which consists of 2 teachers and 4 students.
So, the correct answer is “Option C”.

Note: Candidates must find the number of possibilities of teachers and students are formed from the given data and hence apply the multiplication principle of counting.