Match the counting values. Column I: (A) Arrangements of BANANA with the two Ns not adjacent (B) Words using all letters of IITJEE (C) Number of divisors of 2 2 ⋅ 3 2 ⋅ 5 ⋅ 7 9 (D) If coefficients of

Question

Match the counting values. Column I: (A) Arrangements of BANANA with the two Ns not adjacent (B) Words using all letters of IITJEE (C) Number of divisors of $2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{9}$ (D) If coefficients of the $(r - 1)$ st and $(2 r + 3)$ rd terms of $(1 + x)^{15}$ are equal, find $r$ . Column II: (p) 40 (q) 180 (r) 5 (s) 6

Column I	Column II
(A) Arrangements of BANANA with the two Ns not adjacent	(p) 40
(B) Words using all letters of IITJEE	(q) 180
(C) Number of divisors of $2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{9}$	(r) 5
(D) If coefficients of the $(r - 1)$ st and $(2 r + 3)$ rd terms of $(1 + x)^{15}$ are equal, find $r$ .	(s) 6

TestHub · Accepted Answer

BANANA has 6 ! / ( 3 ! 2 !) = 60 arrangements; adjacent Ns occur in 5 ! /3 ! = 20 , leaving 40. IITJEE has 6 ! / ( 2 ! 2 !) = 180 words. The divisor count is ( 2 + 1 ) ( 2 + 1 ) ( 1 + 1 ) ( 9 + 1 ) = 180 . The two binomial indices are r − 2 and 2 r + 2 ; they sum to 15 at r = 5 .

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