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 ______ . [2018]

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 ______ . [2018]

Smart Achievers · Accepted Answer

(646)

$\sum_{r = 0}^{n} r {({}^{n}C_{r})}^{2} = n \sum_{r = 0}^{n} {}^{n}C_{r} {}^{n - 1}C_{r - 1}$

$= n \sum_{r = 1}^{n} {}^{n}C_{n - r} {}^{n - 1}C_{r - 1} = n {}^{2 n - 1}C_{n - 1}$

$Now, X = {({}^{10}C_{1})}^{2} + 2 {({}^{10}C_{2})}^{2} + 3 {({}^{10}C_{3})}^{2} + \dots + 10 {({}^{10}C_{10})}^{2}$

$= \sum_{n = 0}^{10} r {({}^{10}C_{r})}^{2} = 10 {}^{19}C_{9}$

$∴$ $\frac{X}{1430} = \frac{1}{143} {}^{19}C_{9} = 646$

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 ______ . [2018]

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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 ______ . [2018]

Ans. (646) ∑r=0nr(Crn)2=n∑r=0nCrnCr-1n-1 =n∑r=1nCn-rnCr-1n-1=nCn-12n-1 Now, X=(C110)2+2(C210)2+3(C310)2+⋯+10(C1010)2 =∑n=010r(Cr10)2=10 C919 ∴ X1430=1143 C919=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 ______ . [2018]