The value of - 15 C 1+2 . 15 C 2-3 . 15 C 3+ ldots -15 . 15 C 15+ 14 C 1+ 14 C 3+ 14 C 5+ ldots .+ 14 C 11 is

Question

The value of - 15 C 1+2 . 15 C 2-3 . 15 C 3+ ldots -15 . 15 C 15+ 14 C 1+ 14 C 3+ 14 C 5+ ldots .+ 14 C 11 is (A) 216-1 (B) 213-14 (C) 214 (D) 213-13

Tardigrade Admin · Accepted Answer

(- 15 C 1+2 15 C 2-3 . 15 C 3+ ldots -15 . 15 C 15) +( 14 C 1+ 14 C 3+ ldots .+ 14 C 11) = displaystyle∑ r =115(-1) r ⋅ r 15 C r +( 14 C 1+ 14 C 3+ ldots+ 14 C 11+ 14 C 13)- 14 C 3 = displaystyle∑ r =115(-1) r 15 ⋅ 14 C r -1+213-14 =15(- 14 C 0+ 14 C 1 ldots- 14 C 14)+213-14 =213-14

Q. The value of $-^{15} C_{1} + 2.^{15} C_{2} - 3.^{15} C_{3} + \dots$ $- 15.^{15} C_{15} +^{14} C_{1} +^{14} C_{3} +^{14} C_{5} + \dots . +^{14} C_{11}$ is

$2^{16} - 1$

$2^{13} - 14$

$2^{14}$

$2^{13} - 13$

Q. The value of −15C1​+2.15C2​−3.15C3​+… −15.15C15​+14C1​+14C3​+14C5​+….+14C11​ is

216−1

213−14

214

213−13

(−15C1​+215C2​−3.15C3​+…−15.15C15​) +(14C1​+14C3​+….+14C11​) =r=1∑15​(−1)r⋅r15Cr​+(14C1​+14C3​+…+14C11​+14C13​)−14C3​ =r=1∑15​(−1)r15⋅14Cr−1​+213−14 =15(−14C0​+14C1​…−14C14​)+213−14 =213−14

Q. The value of $-^{15} C_{1} + 2.^{15} C_{2} - 3.^{15} C_{3} + \dots$ $- 15.^{15} C_{15} +^{14} C_{1} +^{14} C_{3} +^{14} C_{5} + \dots . +^{14} C_{11}$ is

$2^{16} - 1$

$2^{13} - 14$

$2^{14}$

$2^{13} - 13$