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?
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?
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 fo 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 $^{22} C_{7}+^{22} C_{10}.$
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