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

If the students will join the party, then the combination will be,

$C_1={}^{22}{C}_7$

$=170544$

If the students will not join the party, then the combination will be,

$C_2={}^{22}{C}_{10}$

$=646646$

The number of ways that the excursion party can be chosen as,

$C=C_1+C_2$

$=170544+646646$

$=817190$