Q 56 For a positive integer n, if the mean of the binomial coefficients in the expansion of a+b2n-3 - Maths - Relations and Functions

Question

Q 56 For a positive integer n, if the mean of the binomial
coefficients in the expansion of a + b 2 n - 3 is 16, then n is
equal to
(1) 5
(2) 7
(3) 9
(4) 4

Aarushi Mishra · Accepted Answer

By bimnomial theorema+b2n-3=C02n-3 anb0+C12n-3 an-1b1+C22n-3 an-2b2+
+Cn2n-3 a0bnput a=b=11+12n-3=C02n-3 1n10+C12n-3 1n-111+C22n-3 1n-212+
+Cn2n-3 101nC02n-3 +C12n-3 +C22n-3 +
+Cn2n-3 =22n-3 ___________1Mean of binomial coefficient=sum of binomial coefficientsTotal number of binoimial coefficientRemember, total number of terms in biniomial expansion x+ym is m+1, so here will be 2n-3+1 binomial coefficentsMean of binomial coefficient=C02n-3 +C12n-3 +C22n-3 +
+Cn2n-32n-3+1=C02n-3 +C12n-3 +C22n-3 +
+Cn2n-32n-2From equaiton 1=22n-32n-1=22n-4n-1=22n-2n-1=4n-2n-1Given mean=164n-2n-1=42Now try with options, first put n=545-25-1=434=42It satisfiesHence n=5

Q.56. For a positive integer n, if the mean of the binomial coefficients in the expansion of a + b 2 n - 3 is 16, then n is equal to (1) 5 (2) 7 (3) 9 (4) 4

Q.56. For a positive integer n, if the mean of the binomial coefficients in the expansion of ${(a + b)}^{2 n - 3}$ is 16, then n is equal to
(1) 5
(2) 7
(3) 9
(4) 4