Question

A delegation of four students is to be selected from a total of 12 students. In how many ways can the delegation be selected if two particular students wished to be included together only?
A. 255
B. 210
C. 400
D. 450

Answer 1

Hint: Divide the solution into two cases: when both the students are selected and when both the students are not selected. In both the cases, use the formula that if we have to select r things from n total things, we can do this in ${}_{r}^{n}C=\dfrac{n!}{r!\left( n-r \right)!}$ ways and then add the two cases to arrive at the final answer.

Complete step-by-step answer:
In this question, we are given that a delegation of four students is to be selected from a total of 12 students.
We need to find the number of ways the delegation can be selected if two particular students wished to be included together only.
So, we need to select four students from a total of 12 students.
Due to the given condition that two particular students wish to be included together only, we will have two cases.
Case 1: when both of them are selected.
Case 2: when both of them are not selected.
Let us start with case 1: when both of them are selected.
In this case, both these particular students are selected. We can do this by 1 way. Now we have to select 2 more students from the rest 10 students.
We know that if we have to select r things from n total things, we can do this in ${}_{r}^{n}C=\dfrac{n!}{r!\left( n-r \right)!}$ ways.
Using this formula, we can find the number of ways to select 2 students from the remaining 10 students.
${}_{2}^{10}C=\dfrac{10!}{2!8!}=45$
So, the case 1 when both the particular students are selected can be done in 45 ways. …(1)
Now, we will solve case 2: when both of them are not selected.
These 2 particular children are not selected. So, now we have to select 4 children from the remaining 10 children. This can be done in ${}_{4}^{10}C=\dfrac{10!}{4!6!}=210$ ways. …(2) So, from equations (1) and (2), we have the following:
Total number of ways in which a delegation of four students is to be selected from a total of 12 students if two particular students wished to be included together only=45 + 210 = 255 ways.
Hence, the correct option is (a).

Note: In this question, it is very important to divide the solution into two cases. It is also important to know that that if we have to select r things from n total things, we can do this in ${}_{r}^{n}C=\dfrac{n!}{r!\left( n-r \right)!}$ ways.Students may confuse while selecting of 4 students from the group of 12.In this question they asked to find selection of 4 students among 12 such a way that 2 students are wished to be included means they are not confirmed among 4 students whether they are select or not, so we have to add both their presence and absence while selecting from the group.