Question

From a class of 25 students, 10 are to be chosen for an excursion party. There are 3 students who decide that either all of them will join or none of them will join. In how many ways can the excursion party be chosen?

Answer 1

Here, total students are 25 from which 10 are chosen for an excursion party. If the 3 students join the party, then 7 students out of the remaining 22 students have to be chosen, or if the 3 students do not join the party, then 10 students out of remaining 22 students will be chosen.

$\therefore$ Number of ways of selection

= ${}^3{C}_3{\times }^{22}{C}_7{+}^3{C}_0{\times }^{22}{C}_{10}$

= $1\times \frac{22!}{7!5!}+1\times \frac{22!}{10!12!}$

= $\frac{22\times 21\times 20\times 19\times 18\times 17\times 16\times 15!}{7\times 6\times 5\times 4\times 3\times 2\times 1\times 15!}$

= $\frac{22\times 21\times 20\times 19\times 18\times 17\times 16\times 15\times 14\times 13\times 12!}{10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1\times 12!}$

= $170544+646646=817190.$