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

The correct options are A 0 if n is odd C ( 1)^n/2 ^n C_n/2 if n is evenWhen 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 ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XI

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

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

(- 1)^{n}

if n is odd

No worries! We‘ve got your back. Try BYJU‘S free classes today!

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

if n is even

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

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

if n is even

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct options are
A 0 if n is odd
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 $x^{2 m} i n (1 - x^{2})^{2 m}$
$= (- 1)^{m} (^{2 m} C_{m}) = (- 1)^{\frac{n}{2}} (^{n} C_{\frac{n}{2}})$

Suggest Corrections

Similar questions

Q. The value of the expression

C_{0}^{2} - C_{1}^{2} + C_{2}^{2} - . . . . . . + {(- 1)}^{n} \times C_{n}^{2}

Q. The value of expression

C_{0}^{2} - C_{1}^{2} + C_{2}^{2} - . . . . . . {(- 1)}^{n} (n + 1) C_{n}^{2}

Q. Value of the expression

C_{0}^{2} + C_{1}^{2} + C_{2}^{2} + . . . . . + (n + 1) C_{n}^{2}

Q. Evaluate

C_{0}^{2} - C_{1}^{2} + C_{2}^{2} - . . . . . . + (- 1)^{n} C_{n}^{2}

for n = 10 and n = 11

Q. If

{(1 + x)}^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . + C_{n} x^{n}

and '

n

' is odd,then value of

C_{0}^{2} - C_{1}^{2} + C_{2}^{2} + . . . + {(- 1)}^{^{n}} C_{n}^{2}

Join BYJU'S Learning Program

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

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