The value of beginpmatrix50 0 endpmatrix beginpmatrix50 1 endpmatrix+ beginpmatrix50 1 endpmatrix beginpmatrix50 2 endpmatrix+ ldots ldots ldots .+ beginpmatrix50 49 endpmatrix beginpmatrix50 50 endpmatrix is, where n Cr= beginpmatrixn r endpmatrix

Question

The value of beginpmatrix50 0 endpmatrix beginpmatrix50 1 endpmatrix+ beginpmatrix50 1 endpmatrix beginpmatrix50 2 endpmatrix+ ldots ldots ldots .+ beginpmatrix50 49 endpmatrix beginpmatrix50 50 endpmatrix is, where n Cr= beginpmatrixn r endpmatrix (A)  beginpmatrix100 50 endpmatrix (B)  beginpmatrix100 51 endpmatrix (C)  beginpmatrix50 25 endpmatrix (D)  beginpmatrix50 25 endpmatrix2

Tardigrade Admin · Accepted Answer

50 C 0 × 50 C 1+ 50 C 4 × 50 C 2+ ldots ldots ldots . .+ 50 C 49 × 50 C 50 = 50 C 0 × 50 C 49+ 50 C 1 × 50 C 48+ ldots ldots ldots ldots+ 50 C 49 × 50 C 0 = co-eff. of x 49 in (1+ x )100= 100 C 49

Q. The value of $(500) (501) + (501) (502) + \dots\dots\dots . + (5049) (5050)$ is, where $^{n} C_{r} = (n r)$

$(10050)$

$(10051)$

$(5025)$

$(5025)^{2}$

$^{50} C_{0} \times^{50} C_{1} +^{50} C_{4} \times^{50} C_{2} + \dots\dots\dots .. +^{50} C_{49} \times^{50} C_{50}$
$=^{50} C_{0} \times^{50} C_{49} +^{50} C_{1} \times^{50} C_{48} + \dots\dots\dots\dots +^{50} C_{49} \times^{50} C_{0}$
$=$ co-eff. of $x^{49}$ in $(1 + x)^{100} =^{100} C_{49}$

Q. The value of (500​)(501​)+(501​)(502​)+……….+(5049​)(5050​) is, where nCr​=(nr​)

(10050​)

(10051​)

(5025​)

(5025​)2