Let X = ( C 1 10 ) 2 + 2 ( C 2 10 ) 2 + 3 ( C 3 10 ) 2 + ? + 10 ( C 10 10 ) 2 , where C r 10 , r { 1 , 2 , … , 10 } , denote binomial coefficients. Then, the value of 1 1430 X is

Question

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 ______ .

Smart Achievers · Accepted Answer

(646)

$X = \sum_{r = 1}^{10} r {({}^{10}C_{r})}^{2} = \sum_{r = 1}^{10} r \cdot {}^{10}C_{r} \cdot {}^{10}C_{r}$

$= 10 \sum_{r = 1}^{10} {}^{9}C_{r - 1} \cdot {}^{10}C_{10 - r} = 10 \cdot {}^{19}C_{9}$

$∴$ $\frac{X}{1430} = \frac{10 \cdot {}^{19}C_{9}}{1430} = \frac{{}^{19}C_{9}}{143} = 646$

Solution

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 ______ .

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Solution

Q. Let X=(C110)2+2(C210)2+3(C310)2+⋯+10(C1010)2, where Cr10, r∈{1,2,…,10}, denote binomial coefficients. Then, the value of 11430X is ______ .

Ans. (646) X=∑r=110r(Cr10)2=∑r=110r·Cr10·Cr10 =10∑r=110Cr-19·C10-r10=10·C919 ∴ X1430=10·C9191430=C919143=646

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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 ______ .