Match the permutation and counting questions. Column I: (A) Permutations of HINDUSTAN containing none of the consecutive patterns HIN, DUS, TAN. (B) Arrangements of MATHEMATICS in which the two Ms or

Question

Match the permutation and counting questions. Column I: (A) Permutations of HINDUSTAN containing none of the consecutive patterns HIN, DUS, TAN. (B) Arrangements of MATHEMATICS in which the two Ms or the two Ts are adjacent. (C) Pairs of distinct integers from 1 to 100 differing by at most 10. (D) Eight-digit positive integers whose digit sum is odd. Column II: (p) 169194 (q) $9 \cdot 9! /2! ($ r $) (2 100) - (2 90)$ (s) $45 \cdot 1 0^{6}$

Column I	Column II
(A) Permutations of HINDUSTAN containing none of the consecutive patterns HIN, DUS, TAN.	(p) 169194
(B) Arrangements of MATHEMATICS in which the two Ms or the two Ts are adjacent.	(q) $9 \cdot 9! /2!$
(C) Pairs of distinct integers from 1 to 100 differing by at most 10.	(r) $(2 100) - (2 90)$
(D) Eight-digit positive integers whose digit sum is odd.	(s) $45 \cdot 1 0^{6}$

TestHub · Accepted Answer

A is 169194 by inclusion-exclusion over the three prohibited patterns. B is 2 ( 10 ! / ( 2 ! 2 !)) − 9 ! /2 ! = 9 ⋅ 9 ! /2 ! . C is ∑ d = 1 10 ​ ( 100 − d ) = 945 , equal to (r). Among 9 ⋅ 1 0 7 eight-digit integers, pairing integers with opposite last-digit parity yields half, 45 ⋅ 1 0 6 .

Mathematics - Permutation & Combination Question with Solution | TestHub

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