Question

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

Answer 1

We have 2 cases

1) when the group of $3$ decides to join

Then we already have those $3$ and we need to select $7$ more from $22$ and that can be done in ${}^{22}{C}_7$ ways

Other case is

2) When the group of $3$ decides not to join

Then we have to choose $10$ from remaining $22$ and that can be done in ${}^{22}{C}_{10}$ ways

So total number of ways are ${}^{22}{C}_7$ + ${}^{22}{C}_{10}$ ways