A team of 5 is to be chosen from 10 students, among whom are Ravi and Mohan. In how many ways can the team be chosen if Ravi must be in the team and Mohan must not be in the team?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 70
A student who must be in takes one place without any choice; a student who must be out is removed from the list. Only the places still open are chosen from the students still available.
Ravi is in: 1 place filled, 4 places left
Ravi and Mohan are both off the list of choices: \(10-2=8\) students left
Ways \(\displaystyle =\,^8C_4=\frac{8\times7\times6\times5}{4\times3\times2\times1}=70\)
Why the other options are incorrect
Option (1) 126 — Incorrect. This is \(^9C_4\); it keeps Ravi in but still allows Mohan to be picked.
Option (3) 56 — Incorrect. This is \(^8C_5\), teams with neither Ravi nor Mohan; Ravi has to be in the team.
Option (4) 252 — Incorrect. This is \(^{10}C_5\), every team of 5, with no condition on Ravi or Mohan.