Question

A kindergarden 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. ${{}^{25}C}_5-{{}^{24}C}_5$
• B. ${{}^{24}C}_5$
• C. ${{}^{24}C}_4$
• D. none

Answer 1

The correct option is B ${{}^{24}C}_5$

Given that no.of visits by a teacher ${=}^{25}{C}_5$

No.of visits by a kid ${=}^{24}{C}_4$

The number of visits of teacher exceeds that of student by ${=}^{25}{C}_5{-}^{24}{C}_4$

We know that ${}^n{C}_r{+}^n{C}_{r-1}{=}^{n+1}{C}_r$

${\Rightarrow }^n{C}_r{=}^{n+1}{C}_r{-}^n{C}_{r-1}$

Put $n=24;r=5$

${\Rightarrow }^{24}{C}_5{=}^{25}{C}_5{-}^{24}{C}_4$

Thus, The number of visits of teacher exceeds that of student by ${=}^{25}{C}_5{-}^{24}{C}_4{=}^{24}{C}_5$