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

From the class of 25 students, 10 are to be chosen for an excursion party.
Since there are 3 students who decide that either all of them will join or none of them will join, there are two cases.

Case I: All the three students join.
Then, the remaining 7 students can be chosen from the remaining 22 students in $^{22}C_7$ ways.

Case II: None of the three students join.
Then, 10 students can be chosen from the remaining 22 students in $^{22}C_{10}$ ways.
Thus, required number of ways of choosing the excursion party is $^{22}C_7+^{22}C_{10}.$