Match List-I with List-II

Choose the correct answer from the options given below:

(A)-(III), (B)-(IV), (C)-(I), (D)-(II)

The correct answer is Option (3) → (A)-(III), (B)-(IV), (C)-(I), (D)-(II)

(A) $^{75}P_2 - ^{75}C_2$

• $^{75}P_2 = 75 \times 74 = 5550$
• $^{75}C_2 = \frac{75 \times 74}{2 \times 1} = 2775$
• Difference: $5550 - 2775 = \mathbf{2775}$
• Matches: (III)

(B) $^5P_5 - ^{10}C_3$

• $^5P_5 = 5! = 120$
• $^{10}C_3 = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120$
• Difference: $120 - 120 = \mathbf{0}$
• Matches: (IV)

(C) $^{16}C_{13} - ^8C_3$

• $^{16}C_{13} = ^{16}C_{16-13} = ^{16}C_3 = \frac{16 \times 15 \times 14}{3 \times 2 \times 1} = 560$
• $^8C_3 = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56$
• Difference: $560 - 56 = \mathbf{504}$
• Matches: (I)

(D) $^nP_4 = 360$

• We need to find $n$ such that $n(n-1)(n-2)(n-3) = 360$.
• Testing $n = 6$: $6 \times 5 \times 4 \times 3 = 360$.
• Matches: (II)