Let X = (10C1)2+2(10C2)2+3(10C3)2+⋅s+10(10C10)2, where 10Cr, r ∈ 1, 2, ⋅s, 10 denote binomial coefficients. Then, the value of (1/1430) X is .

Question

Let X = (10C1)2+2(10C2)2+3(10C3)2+⋅s+10(10C10)2, where 10Cr, r ∈ 1, 2, ⋅s, 10 denote binomial coefficients. Then, the value of (1/1430) X is . (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

X= displaystyle∑r=0n r ⋅( n Cr)2 ; n=10 X=n ⋅ displaystyle∑r=0n n Cr ⋅ n-1 Cr-1 X=n ⋅ displaystyle∑r=0n n Cn-r ⋅ n-1 Cr-1 X=n ⋅ 2 n-1 Cn-1 ; n=10 X=10 . 19 C9 (X/1430)=(1/143) ⋅ 19 C9=646

Q. Let $X = (^{10} C_{1})^{2} + 2 (^{10} C_{2})^{2} + 3 (^{10} C_{3})^{2} + \dots + 10 (^{10} C_{10})^{2}$ ,
where $^{10} C_{r}, r \in {1, 2, \dots, 10}$ denote binomial coefficients. Then, the value of $\frac{1}{1430}$ X is _____ .

$X = r = 0 \sum n r \cdot (^{n} C_{r})^{2}; n = 10$
$X = n \cdot r = 0 \sum n^{n} C_{r} \cdot^{n - 1} C_{r - 1}$
$X = n \cdot r = 0 \sum n^{n} C_{n - r} \cdot^{n - 1} C_{r - 1}$
$X = n \cdot^{2 n - 1} C_{n - 1}; n = 10$
$X = 10.^{19} C_{9}$
$\frac{X}{1430} = \frac{1}{143} \cdot^{19} C_{9} = 646$

Q. Let X=(10C1​)2+2(10C2​)2+3(10C3​)2+⋯+10(10C10​)2, where 10Cr​,r∈{1,2,⋯,10} denote binomial coefficients. Then, the value of 14301​ X is _____ .

X=r=0∑n​r⋅(nCr​)2;n=10 X=n⋅r=0∑n​nCr​⋅n−1Cr−1​ X=n⋅r=0∑n​nCn−r​⋅n−1Cr−1​ X=n⋅2n−1Cn−1​;n=10 X=10.19C9​ 1430X​=1431​⋅19C9​=646

Q. Let $X = (^{10} C_{1})^{2} + 2 (^{10} C_{2})^{2} + 3 (^{10} C_{3})^{2} + \dots + 10 (^{10} C_{10})^{2}$ ,
where $^{10} C_{r}, r \in {1, 2, \dots, 10}$ denote binomial coefficients. Then, the value of $\frac{1}{1430}$ X is _____ .

$X = r = 0 \sum n r \cdot (^{n} C_{r})^{2}; n = 10$
$X = n \cdot r = 0 \sum n^{n} C_{r} \cdot^{n - 1} C_{r - 1}$
$X = n \cdot r = 0 \sum n^{n} C_{n - r} \cdot^{n - 1} C_{r - 1}$
$X = n \cdot^{2 n - 1} C_{n - 1}; n = 10$
$X = 10.^{19} C_{9}$
$\frac{X}{1430} = \frac{1}{143} \cdot^{19} C_{9} = 646$