Match List-I with List-II.
| List-I | List-II |
| (A) \(^6P_2\) | (I) 9 |
| (B) \(^7C_3\) | (II) 12 |
| (C) The number of diagonals of a hexagon | (III) 30 |
| (D) The number of ways in which all the letters of the word COOK can be arranged (each letter used once, the two O’s identical) | (IV) 35 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
\(^nP_r\) 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) \(^6P_2\) — this matches (III) 30.
\(^6P_2=6\times5=30\)
(B) \(^7C_3\) — this matches (IV) 35.
\(\displaystyle ^7C_3=\frac{7\times6\times5}{3\times2\times1}=35\)
(C) The number of diagonals of a hexagon — this matches (I) 9.
Joining any 2 of the 6 corners gives a line: \(\displaystyle ^6C_2=\frac{6\times5}{2}=15\)
6 of these lines are the sides, the rest are diagonals: \(15-6=9\)
(D) The number of ways in which all the letters of the word COOK can be arranged — this matches (II) 12.
COOK has 4 letters with O twice
Arrangements \(\displaystyle =\frac{4!}{2!}=\frac{24}{2}=12\)