Match the binomial and counting results. Column I: (A) In ( 2 x + 4 − x ) n , the second-to-third term ratio is 1/7 and the sum of their coefficients is 36. Find x . (B) The coefficients of the second

Question

Match the binomial and counting results. Column I: (A) $In (2^{x} + 4^{- x})^{n}$ , the second-to-third term ratio is $1/7$ and the sum of their coefficients is 36. Find $x$ . (B) The coefficients of the second, third and fourth terms of $(1 + x)^{n}$ are in arithmetic progression. Find $n$ . (C) A basket has four oranges, five apples and six mangoes. Find the ways to choose at least one fruit. (D) If $(3 r 31) = (r + 3 31)$ , find $r$ . Column II: (p) 1/3 (q) 7 (r) -1/3 (s) 209

Column I	Column II
(A) $In (2^{x} + 4^{- x})^{n}$ , the second-to-third term ratio is $1/7$ and the sum of their coefficients is 36. Find $x$ .	(p) 1/3
(B) The coefficients of the second, third and fourth terms of $(1 + x)^{n}$ are in arithmetic progression. Find $n$ .	(q) 7
(C) A basket has four oranges, five apples and six mangoes. Find the ways to choose at least one fruit.	(r) -1/3
(D) If $(3 r 31) = (r + 3 31)$ , find $r$ .	(s) 209

TestHub · Accepted Answer

In A , n + ( 2 n ​ ) = 36 gives n = 8 ; the term ratio then gives x = − 1/3 . In B , ( 1 n ​ ) + ( 3 n ​ ) = 2 ( 2 n ​ ) gives n = 7 (for n ≥ 3 ). In C, treating fruit of each type as alike, there are 5 ⋅ 6 ⋅ 7 − 1 = 209 selections. In D, symmetry of the coefficients gives 3 r + ( r + 3 ) = 31 , hence r = 7 .

Mathematics - Binomial Theorem Question with Solution | TestHub

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