The value of displaystyle∑m=02013 displaystyle∑n=0m beginpmatrix2013 m endpmatrix beginpmatrixm n endpmatrix equals

Question

The value of displaystyle∑m=02013 displaystyle∑n=0m beginpmatrix2013 m endpmatrix beginpmatrixm n endpmatrix equals (A) 2014 (B) 22014 (C) 32013 (D) 2013

Tardigrade Admin · Accepted Answer

We have displaystyle∑ m =02013( 2013 C m )[ m C 0+ m C 1+ ldots ldots .+ m C m ](. opening . displaystyle∑ n =0 m ) = displaystyle∑ m =02013 2013 C m ⋅ 2 m = 2013 C 0+ 2013 C 1 ⋅ 2+ ldots ldots . .+ 2013 C 2013 ⋅ 22013=(1+2)2013=32013

Q. The value of $m = 0 \sum 2013 n = 0 \sum m (2013 m) (m n)$ equals

2014

$2^{2014}$

$3^{2013}$

2013

We have $m = 0 \sum 2013 (^{2013} C_{m}) [^{m} C_{0} +^{m} C_{1} + \dots\dots . +^{m} C_{m}] ($ opening $n = 0 \sum m)$
$= m = 0 \sum 2013^{2013} C_{m} \cdot 2^{m} =^{2013} C_{0} +^{2013} C_{1} \cdot 2 + \dots\dots .. +^{2013} C_{2013} \cdot 2^{2013} = (1 + 2)^{2013} = 3^{2013}$

Q. The value of m=0∑2013​n=0∑m​(2013m​)(mn​) equals

2014

22014

32013

2013

We have m=0∑2013​(2013Cm​)[mC0​+mC1​+…….+mCm​]( opening n=0∑m​) =m=0∑2013​2013Cm​⋅2m=2013C0​+2013C1​⋅2+……..+2013C2013​⋅22013=(1+2)2013=32013

Q. The value of $m = 0 \sum 2013 n = 0 \sum m (2013 m) (m n)$ equals

$2^{2014}$

$3^{2013}$

We have $m = 0 \sum 2013 (^{2013} C_{m}) [^{m} C_{0} +^{m} C_{1} + \dots\dots . +^{m} C_{m}] ($ opening $n = 0 \sum m)$
$= m = 0 \sum 2013^{2013} C_{m} \cdot 2^{m} =^{2013} C_{0} +^{2013} C_{1} \cdot 2 + \dots\dots .. +^{2013} C_{2013} \cdot 2^{2013} = (1 + 2)^{2013} = 3^{2013}$