Question

The kindergarten teacher has 25 kids in her class. She takes 5 of them at a time, to zoological garden as often as she can, without taking the same 5 kids more than once; then the number of visits, the teacher makes to the garden exceeds that of a kid by:
(a) ${}^{25}{{C}_{5}}-{}^{24}{{C}_{4}}$
(b) ${}^{24}{{C}_{7}}$
(c) ${}^{25}{{C}_{5}}-{}^{24}{{C}_{5}}$
(d) ${}^{24}{{C}_{4}}$

Answer 1

Hint: Find the total number of ways to select 5 kids out of 25 by using the formula: ${}^{n}{{C}_{r}}$, where ‘n’ is the total number of kids and ‘r’ is the number of kids to be selected. This will be the total number of times the teacher will make the visit to the zoological garden. To determine the total number of visits of one kid, take that kid beside from the group and find the total number of ways the remaining 24 kids can be formed in a group of 4 by using the relation: ${}^{n-1}{{C}_{r-1}}$. This will be the total number of visits one kid will make. Take the difference of the two expressions to get the answer.

Complete step-by-step answer:
It is given that, at a time the teacher takes 5 kids out of 25 kids to the zoological garden. Therefore, the number of ways to select 5 kids out of 25 = ${}^{25}{{C}_{5}}$. So, the number of times the teacher will visit the zoological garden = ${}^{25}{{C}_{5}}$.
Now, it is given that the same five kids are not taken more than once. Therefore, we take one of the kids beside us from the group.
Let us select 4 kids from the remaining 24 kids and the last seat remains vacant for the kid who has been taken beside such that every time a new group is formed.
So, number of ways to select 4 kids out of 24 kids = ${}^{n-1}{{C}_{r-1}}={}^{24}{{C}_{4}}$. Therefore, the number of times one kid will make a visit = ${}^{24}{{C}_{4}}$.
Therefore, we have got that the number of visits the teacher makes to the garden exceeds that of a kid by ${}^{25}{{C}_{5}}-{}^{24}{{C}_{4}}$.
Hence, option (a) is the correct answer.

Note: One may note that we have taken a kid beside from the group. This is done because we have to determine the number of times one kid can make a visit to the zoological garden. So we have arranged groups of 4 kids among 24 kids. In doing so, one place remains vacant for the fifth kid and we can arrange him in each group so that every time we can form a group of 5 different kids. Since, the teacher will visit the garden each time, so she has to visit the garden the number of times different groups of 5 kids can be formed which is ${}^{25}{{C}_{5}}$.