Question

Match the entries of List-I with the correct entries of List-II.

List-I

(P) The number of positive Integral solution(s) of $x_1x_2x_3 = 360$ Is (Q) If S = {1, 2, 3, 4, ..., 9}. Then the number of subsets of S containing atleast one odd number Is (R) The number of five digit numbers In which every digit exceeds the Immediately preceding digit (when read from left to right) is (S) The number of positive Integral solution(s) of $x_1 + x_2 + x_3 + x_4 = 12$ Is

List-II

(1) 165 (2) 455 (3) 126 (4) 180 (5) 496

Then the correct option is

• A. (P) $\rightarrow$ (5); (Q) $\rightarrow$ (4); (R) $\rightarrow$ (3); (S) $\rightarrow$ (1)
• B. (P) $\rightarrow$ (3); (Q) $\rightarrow$ (5); (R) $\rightarrow$ (3); (S) $\rightarrow$ (2)
• C. (P) $\rightarrow$ (4); (Q) $\rightarrow$ (5); (R) $\rightarrow$ (3); (S) $\rightarrow$ (1)
• D. (P) $\rightarrow$ (4); (Q) $\rightarrow$ (3); (R) $\rightarrow$ (1); (S) $\rightarrow$ (2)

Answer 1

Answer: (C)

(P) $\rightarrow$ (4); (Q) $\rightarrow$ (5); (R) $\rightarrow$ (3); (S) $\rightarrow$ (1)

Answer 2

(P) $360 = 2^3 \times 3^2 \times 5^1$. Number of solutions is $\binom{3+3-1}{3-1}\binom{2+3-1}{3-1}\binom{1+3-1}{3-1} = \binom{5}{2}\binom{4}{2}\binom{3}{2} = 10 \times 6 \times 3 = 180$. (Q) Total subsets of S (9 elements) is $2^9 = 512$. Odd numbers are {1, 3, 5, 7, 9} (5). Even numbers are {2, 4, 6, 8} (4). Subsets with no odd numbers = $2^4 = 16$. Subsets with at least one odd number = $512 - 16 = 496$. (R) Choose 5 distinct digits from {1, 2, ..., 9}. $\binom{9}{5} = 126$. (S) Using stars and bars for positive integers: $\binom{12-1}{4-1} = \binom{11}{3} = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} = 165$.