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?

Question

Number of seven digits numbers which can be formed by using all the digits 1, 2, 3, 4, 5 with at least one digit repeated is _____.

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 \(^{22}C_7+^{22}C_{10}.\)

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?

Solution and Explanation

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