Let S =30 ⋅ 15 C 0+27 ⋅ 15 C 1+24 ⋅ 15 C 2+ ldots ldots . .12 ⋅ 15 C 14-15 ⋅ 15 C 15 has the value α ⋅ 2β, where α, β ∈ N then the least value of |α-β| is equal to

Question

Let S =30 ⋅ 15 C 0+27 ⋅ 15 C 1+24 ⋅ 15 C 2+ ldots ldots . .12 ⋅ 15 C 14-15 ⋅ 15 C 15 has the value α ⋅ 2β, where α, β ∈ N then the least value of |α-β| is equal to (A) 1 (B) 2 (C) 4 (D) none

Tardigrade Admin · Accepted Answer

S =30 ⋅ C 0 ldots ldots-15 ⋅ C 15 S = -15 C 0 ldots ldots+30 ⋅ C 15 2 S =15( C 0+ ldots ldots+ C 15)=15 ⋅ 215 ∴ S =15 ⋅ 214 ∴ α=15 ; β=14

Q. Let $S = 30 \cdot^{15} C_{0} + 27 \cdot^{15} C_{1} + 24 \cdot^{15} C_{2} + \dots\dots ..12 \cdot^{15} C_{14} - 15 \cdot^{15} C_{15}$ has the value $α \cdot 2^{β}$ , where $α, β \in N$ then the least value of $∣ α - β ∣$ is equal to

1

2

4

none

$S = 30 \cdot C_{0} \dots\dots - 15 \cdot C_{15}$
$S =^{- 15} C_{0} \dots\dots + 30 \cdot C_{15}$
__________________________
$2 S = 15 (C_{0} + \dots\dots + C_{15}) = 15 \cdot 2^{15}$
$∴ S = 15 \cdot 2^{14}$
$∴ α = 15; β = 14$

Q. Let S=30⋅15C0​+27⋅15C1​+24⋅15C2​+……..12⋅15C14​−15⋅15C15​ has the value α⋅2β, where α,β∈N then the least value of ∣α−β∣ is equal to

1

2

4

none

S=30⋅C0​……−15⋅C15​ S=−15C0​……+30⋅C15​ __________________________ 2S=15(C0​+……+C15​)=15⋅215 ∴S=15⋅214 ∴α=15;β=14

Q. Let $S = 30 \cdot^{15} C_{0} + 27 \cdot^{15} C_{1} + 24 \cdot^{15} C_{2} + \dots\dots ..12 \cdot^{15} C_{14} - 15 \cdot^{15} C_{15}$ has the value $α \cdot 2^{β}$ , where $α, β \in N$ then the least value of $∣ α - β ∣$ is equal to

$S = 30 \cdot C_{0} \dots\dots - 15 \cdot C_{15}$
$S =^{- 15} C_{0} \dots\dots + 30 \cdot C_{15}$
__________________________
$2 S = 15 (C_{0} + \dots\dots + C_{15}) = 15 \cdot 2^{15}$
$∴ S = 15 \cdot 2^{14}$
$∴ α = 15; β = 14$