If Cn Cn-2 - Cn-1Cn-3 - Cn-2Cn-4 - - C3C1 - C2C0 - 2nCn- - where Ci - n Ci - N- then find the value of -

C_0 C_2 + C_1C_3 + C_2C_4 + ..........+ C_n 2C_n = ^2nC_n+ 2 nbsp; (1 + x)^n = ^n C_0 + ^n C_1x + ^n C_2 x^2 + ^n C_3x^3 + ......+ ^n C_nx^n ....... ...

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Byju's Answer

Standard XII

Mathematics

Discriminant

If Cn Cn-2 ...

Question

If $C_{n} C_{n - 2} + C_{n - 1} C_{n - 3} + C_{n - 2} C_{n - 4} + . . . + C_{3} C_{1} + C_{2} C_{0} =^{2 n} C_{n - λ}$ where $C_{i} =^{n} C_{i}, λ \in N$ , then find the value of $λ$

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Solution

$C_{0} C_{2} + C_{1} C_{3} + C_{2} C_{4} + . . . . . . . . . . + C_{n - 2} C_{n} =^{2 n} C_{n + 2}$
$(1 + x)^{n} =^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} +^{n} C_{3} x^{3} + . . . . . . +^{n} C_{n} x^{n}$ ........(1)
$(x + 1)^{n} =^{n} C_{0} x^{n} +^{n} C_{1} x^{n - 1} +^{n} C_{2} x^{n - 2} + . . . . . . +^{n} C_{n}$ ........(2)
multiplying equation (1) & (2)
$(1 + x)^{2 n} = (^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} + . . . . +^{n} C_{n} x^{n}) (^{n} C_{0} x^{n} +^{n} C_{1} x^{n - 1} +^{n} C_{2} x^{n - 2} + . . . . . . +^{n} C_{n})$ .
Now required expression is coefficient of $x^{n - 2} i n (1 + x)^{2 n} =^{2 n} C_{n - 2} =^{2 n} C_{n + 2}$

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Similar questions

C_{0} C_{2} + C_{1} C_{3} + C_{2} C_{4} + . . . + C_{n - 2} C_{n} =^{2 n} C_{n - 1}

^{2 n} C_{n + 1}

Q. If

∣ ∣ ∣ ∣ \begin{matrix} λ^{2} + 3 λ & λ - 1 & λ + 3 λ + 1 & 1 - 2 λ & λ - 4 λ - 2 & λ + 4 & 3 λ \end{matrix} ∣ ∣ ∣ ∣ = p λ^{4} + q λ^{3} + r λ^{2} + s λ + t

be an identity in

λ

, where p, q, r, s and t are constants, find the value of t.

Q. Let

p λ^{4} + q λ^{3} + r λ^{2} + s λ + t = ∣ ∣ ∣ ∣ ∣ \begin{matrix} λ^{2} + 3 λ & λ - 1 & λ + 3 λ + 1 & - 2 λ & λ - 4 λ - 3 & λ + 4 & 3 λ \end{matrix} ∣ ∣ ∣ ∣ ∣

be an identity in

λ

, where

p, q, r, s, t

are constants. Then the value of

t

is ...............

Q. If

| a | = 3, | b | = 4

then find the value of

λ

for which

a + λ b

is perpendicular to

a - λ b

Q. Let

[\sqrt{n^{2} + 1}] = [\sqrt{n^{2} + λ}]

, where

n, λ \in N

. Then

λ

can have

y \times n

different values. Find

y

(Note :

[.]

represents the greatest integer function.)

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If CnCn−2+Cn−1Cn−3+Cn−2Cn−4+...+C3C1+C2C0=2nCn−λ where Ci=nCi,λ∈N, then find the value of λ

If $C_{n} C_{n - 2} + C_{n - 1} C_{n - 3} + C_{n - 2} C_{n - 4} + . . . + C_{3} C_{1} + C_{2} C_{0} =^{2 n} C_{n - λ}$ where $C_{i} =^{n} C_{i}, λ \in N$ , then find the value of $λ$