For r = 0 , 1 , … , 10 , let A r , B r and C r denote, respectively, the coefficient of x r in the expansions of ( 1 + x ) 10 , ( 1 + x ) 20 and ( 1 + x ) 30 . Then r = 1 10 A r ( B 10 B r - C 10 A r ) is equal to [2010]

Question

For $r = 0, 1, \dots, 10,$ let $A_{r}, B_{r}$ and $C_{r}$ denote, respectively, the coefficient of $x^{r}$ in the expansions of ${(1 + x)}^{10}, {(1 + x)}^{20}$ and ${(1 + x)}^{30}$ . Then $\sum_{r = 1}^{10} A_{r} (B_{10} B_{r} - C_{10} A_{r})$ is equal to [2010]

Smart Achievers · Accepted Answer

(4)

$Clearly A_{r} = {}^{10}C_{r}, B_{r} = {}^{20}C_{r}, C_{r} = {}^{30}C_{r}$

Now $\sum_{r = 1}^{10} A_{r} (B_{10} B_{r} - C_{10} A_{r})$

$= \sum_{r = 1}^{10} {}^{10}C_{r} ({}^{20}C_{10} {}^{20}C_{r} - {}^{30}C_{10} {}^{30}C_{r})$

$= {}^{20}C_{10} \sum_{r = 1}^{10} {}^{10}C_{r} {}^{20}C_{r} - {}^{30}C_{10} \sum_{r = 1}^{10} {}^{10}C_{r} \cdot {}^{10}C_{r}$

$= {}^{20}C_{10} ({}^{10}C_{1} {}^{20}C_{1} + {}^{10}C_{2} {}^{20}C_{2} + \dots + {}^{10}C_{10} {}^{20}C_{10})$ $- {}^{30}C_{10} ({}^{10}C_{1} \times {}^{10}C_{1} + {}^{10}C_{2} \times {}^{10}C_{2} + \dots + {}^{10}C_{10} \times {}^{10}C_{10}) ...(i)$

Now, on expanding ${(1 + x)}^{10}$ and ${(1 + x)}^{20}$ and comparing coefficients of $x^{20}$ in their product on both sides, we get

${}^{10}C_{0} {}^{20}C_{0} + {}^{10}C_{1} {}^{20}C_{1} + {}^{10}C_{2} {}^{20}C_{2} + \dots + {}^{10}C_{10} {}^{20}C_{10}$

$= Coeff. of x^{20} in {(1 + x)}^{30}$ $= {}^{30}C_{20} = {}^{30}C_{10}$

$∴$ ${}^{10}C_{1} {}^{20}C_{1} + {}^{10}C_{2} {}^{20}C_{2} + \dots + {}^{10}C_{10} {}^{20}C_{10} = {}^{30}C_{10} - 1$ $. . . (i i)$

$Again on expanding {(1 + x)}^{10} and {(x + 1)}^{10} and comparing the coefficients of x^{10} in their product on both sides, we get$

$∴$ ${({}^{10}C_{0})}^{2} + {({}^{10}C_{1})}^{2} + \dots + {({}^{10}C_{10})}^{2}$

$= Coeff. of x^{10} in {(1 + x)}^{20}$ $= {}^{20}C_{10}$

$∴$ ${({}^{10}C_{1})}^{2} + {({}^{10}C_{2})}^{2} + \dots + {({}^{10}C_{10})}^{2} = {}^{20}C_{10} - 1$ $. . . (i i i)$

Now from equations (i), (ii), and (iii), we get

$Required value = {}^{20}C_{10} ({}^{30}C_{10} - 1) - {}^{30}C_{10} ({}^{20}C_{10} - 1)$

$= {}^{30}C_{10} - {}^{20}C_{10} = C_{10} - B_{10}$

Solution

Q.
For $r = 0, 1, \dots, 10,$ let $A_{r}, B_{r}$ and $C_{r}$ denote, respectively, the coefficient of $x^{r}$ in the expansions of ${(1 + x)}^{10}, {(1 + x)}^{20}$ and ${(1 + x)}^{30}$ . Then $\sum_{r = 1}^{10} A_{r} (B_{10} B_{r} - C_{10} A_{r})$ is equal to [2010]

1 $B_{10} - C_{10}$

2 $A_{10} ({B^{2}}_{10} C_{10} A_{10})$

3 $0$

4 $C_{10} - B_{10}$

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Solution

Q. For r=0,1,…,10, let Ar,Br and Cr denote, respectively, the coefficient of xr in the expansions of (1+x)10,(1+x)20 and (1+x)30. Then ∑r=110Ar(B10Br-C10Ar) is equal to [2010]

1 B10-C10

2 A10(B210C10A10)

3 0