Match List-I with List-II.
| List-I | List-II |
| (A) \(^7P_2\) | (I) 6 |
| (B) \(^8C_3\) | (II) 21 |
| (C) The number of straight lines through 2 of 7 points, no three of which lie on one line | (III) 42 |
| (D) The number of ways in which all the letters of the word NOON can be arranged (each letter used once, the two N’s identical and the two O’s identical) | (IV) 56 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A)-(III), (B)-(IV), (C)-(II), (D)-(I)
\(^nP_r\) counts arrangements, the product of \(r\) numbers counting down from \(n\); \(^nC_r\) counts selections, \(\frac{n!}{r!\,(n-r)!}\). Each List-I item is worked out with these.
(A) \(^7P_2\) — this matches (III) 42.
\(^7P_2=7\times6=42\)
(B) \(^8C_3\) — this matches (IV) 56.
\(\displaystyle ^8C_3=\frac{8\times7\times6}{3\times2\times1}=56\)
(C) The number of straight lines through 2 of 7 points, no three of which lie on one line — this matches (II) 21.
Every pair of points gives one line: \(\displaystyle ^7C_2=\frac{7\times6}{2}=21\)
(D) The number of ways in which all the letters of the word NOON can be arranged — this matches (I) 6.
NOON has 4 letters: N twice and O twice
Arrangements \(\displaystyle =\frac{4!}{2!\times2!}=\frac{24}{4}=6\)