Read the statements given below carefully.
(A) The value of \(^{10}C_7\) is 120.
(B) Five different people, one of whom is Mohan, stand in a single queue. The number of ways in which they can stand with Mohan at the front of the queue is 120.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Only (A)
\(^nC_r=\frac{n!}{r!\,(n-r)!}\), and choosing \(r\) things is the same as leaving out \(n-r\), so \(^nC_r=\,^nC_{n-r}\). A place with a condition is filled first, then the rest are arranged.
(A) The value of \(^{10}C_7\) is 120 Correct.
\(^{10}C_7=\,^{10}C_3\)
\(\displaystyle ^{10}C_3=\frac{10\times9\times8}{3\times2\times1}=120\)
(B) The number of ways in which they can stand with Mohan at the front of the queue is 120 Incorrect.
Mohan is fixed at the front: 1 way
The other 4 people fill the 4 places behind him: \(4!=24\) ways
Ways \(=24\), not 120 (120 is \(5!\), the count with no condition)