The value of the expression C02-C12-C22- -1n Cn2

When n is odd, taken n = 2m + 1, so that S=C_0^2 C_1^2+C_2^2 ⋯ +( 1)^2m C_2m^2+( 1)^2m+1 C_2m+1^2 =(C_0^2 C_2m+1^2 ) (C_1^2 C_2m^2 )+ ⋯ But ^2m+1C_0 = ^2m+1C ...

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Byju's Answer

Standard XIII

Mathematics

Sum of Product of Binomial Coefficients

The value of ...

Question

The value of the expression $C_{0}^{2} - C_{1}^{2} + C_{2}^{2} - \dots + (- 1)^{n} C_{n}^{2}$

0 if n is odd

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(- 1)^{n}

if n is odd

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(- 1)^{\frac{n}{2}}^{n} C_{\frac{n}{2}}

if n is even

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(- 1)^{n - 1}^{n} C_{n - 1}

if n is even

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Solution

The correct option is C $(- 1)^{\frac{n}{2}}^{n} C_{\frac{n}{2}}$ if n is even
When n is odd, taken n = 2m + 1, so that
$S = C_{0}^{2} - C_{1}^{2} + C_{2}^{2} - \dots + (- 1)^{2 m} C_{2 m}^{2} + (- 1)^{2 m + 1} C_{2 m + 1}^{2}$
$= (C_{0}^{2} - C_{2 m + 1}^{2}) - (C_{1}^{2} - C_{2 m}^{2}) + \dots$
But $^{2 m + 1} C_{0} =^{2 m + 1} C_{2 m + 1},$
$^{2 m + 1} C_{1} =^{2 m + 1} C_{2 m}$ etc.
Therefore S = 0
When n is even, we take n = 2m. In this case
$C_{0}^{2} - C_{1}^{2} + C_{2}^{2} - C_{3}^{2} + \dots + (- 1)^{2 m} C_{2 m}^{2}$
= Coefficient of constant term in
$[C_{0} - C_{1} x + C_{2} x^{2} - \dots + (- 1)^{2 m} C_{2 m} x^{2 m}]$
$[C_{0} + C_{1} \frac{1}{x} + c_{2} \frac{1}{x^{2}} + \dots + C_{2 m} \frac{1}{x^{2 m}}]$
= Coefficient of constant term in
$(1 - x)^{2 m} {(1 + \frac{1}{x})}^{2 m}$
= Coefficient of $x^{2 m} i n (1 - x)^{2 m} (1 + x)^{2 m}$
= Coefficient of $x^{2 m} i n (1 - x^{2})^{2 m}$
$= (- 1)^{m} (^{2 m} C_{m}) = (- 1)^{\frac{n}{2}} (^{n} C_{\frac{n}{2}})$

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The value of the expression C20−C21+C22−⋯+(−1)nC2n

The value of the expression $C_{0}^{2} - C_{1}^{2} + C_{2}^{2} - \dots + (- 1)^{n} C_{n}^{2}$