The value of ( 30 0 ) ( 30 10 ) - ( 30 1 ) ( 30 11 ) + ( 30 2 ) ( 30 12 ) - ? + ( 30 20 ) ( 30 30 ) is where ( n r ) = C r n [2005]

Question

The value of $(\binom{30}{0}) (\binom{30}{10}) - (\binom{30}{1}) (\binom{30}{11}) + (\binom{30}{2}) (\binom{30}{12}) - \dots + (\binom{30}{20}) (\binom{30}{30})$ is

where $(\binom{n}{r}) = {}^{n}C_{r}$ [2005]

Smart Achievers · Accepted Answer

(1)

$To find {}^{30}C_{0} {}^{30}C_{10} - {}^{30}C_{1} {}^{30}C_{11} + {}^{30}C_{2} {}^{30}C_{12} - \dots + {}^{30}C_{20} {}^{30}C_{30}$

$∵$ ${(1 + x)}^{30} = {}^{30}C_{0} + {}^{30}C_{1} x + {}^{30}C_{2} x^{2} + \dots + {}^{30}C_{20} x^{20} + . . . + {}^{30}C_{30} x^{30} . . . (i)$

$and {(x - 1)}^{30} = {}^{30}C_{0} x^{30} - {}^{30}C_{1} x^{29} + \dots + {}^{30}C_{10} x^{20} - {}^{30}C_{11} x^{19} + {}^{30}C_{12} x^{18} + \dots + {}^{30}C_{30} x^{0} . . . (i i)$

$On multiplying (i) and (ii), we get$

${(x^{2} - 1)}^{30} = {(1 + x)}^{30} {(x - 1)}^{30}$

$Equating the coefficients of x^{20} on both sides, we get$

${}^{30}C_{10} = {}^{30}C_{0} {}^{30}C_{10} - {}^{30}C_{1} {}^{30}C_{11} + {}^{30}C_{2} {}^{30}C_{12} - \dots + {}^{30}C_{20} {}^{30}C_{30}$

$∴$ $Required value = {}^{30}C_{10}$

Solution

Q.
The value of $(\binom{30}{0}) (\binom{30}{10}) - (\binom{30}{1}) (\binom{30}{11}) + (\binom{30}{2}) (\binom{30}{12}) - \dots + (\binom{30}{20}) (\binom{30}{30})$ is

where $(\binom{n}{r}) = {}^{n}C_{r}$ [2005]

1 $(\binom{30}{10})$

2 $(\binom{30}{15})$

3 $(\binom{60}{30})$

4 $(\binom{31}{10})$

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Solution

Q. The value of (300)(3010)-(301)(3011)+(302)(3012)-⋯+(3020)(3030) is where (nr)=Crn [2005]

1 (3010)

2 (3015)

3 (6030)