Find the value of C 0 C 3- C 1 C 4- C n -3 C n- when C r - n C rA- 2 n CnB- 2 n Cn-2C- 2 n Cn-3D- 2 n Cn-4

The correct option is C We know ^nC_3 = ^nC_n 3, ^nC_4 = ^nC_n 4 . . . . . ⇒ sum = ^nC_0 ^nC_n 3 + ^nC_1 ^nC_n 4 . . . . ^nC_n 3^nC_0 The sum of the ...

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Byju's Answer

Standard XI

Mathematics

Binomial Coefficients

Find the valu...

Question

Find the value of $C_{0} C_{3} + C_{1} C_{4} + . . . . . C_{n - 3} C_{n}$ , when $C_{r} =^{n} C_{r}$

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Solution

The correct option is A

We know $^{n} C_{3} =^{n} C_{n} - 3,^{n} C_{4} =^{n} C_{n} - 4 . . . . .$
$\Rightarrow s u m =^{n} C_{0}^{n} C_{n - 3} +^{n} C_{1}^{n} C_{n - 4} . . . .^{n} C_{n - 3}^{n} C_{0}$
The sum of the subscripts is same in each term = n - 3
Consider the expansion of $(1 + x)^{n} (1 + x)^{n}$ and find the coefficient of $x^{n - 3}$ .
$(1 + x)^{n} (1 + x)^{n} = (^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} . . . . .^{n} C_{n} x^{n}) (^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x_{2} . . . . .^{n} C_{n} x^{n})$
Coefficient of $x^{n - 3} =^{n} C_{0} \times^{n} C n - 3 +^{n} C_{1}^{n} C_{n - 4} . . . . .^{n} C_{n - 3}^{n} C_{0}$
This is same as the required. But we can calculate the coefficient of $x^{n - 3}$ easily.
Sum = coefficient of $x^{n - 3} i n (1 + x)^{n} (1 + x)^{n}$
= coefficient of $x^{n - 3} i n (1 + x)^{2 n}$
= $^{2 n} C_{n - 3}$

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Similar questions

Find the value of $C_{0}^{2} + C_{1}^{2} + C_{2}^{2} . . . . . C_{n}^{2}$ , where $C_{r} =^{n} C_{r}$

The middle term in the expansion of $(x^{2} + \frac{1}{x^{2}} + 2)^{n}$ is:

Q. For

r = 0, 1, 2, . . . ., n

, prove that

C_{0} \cdot C_{r} + C_{1} \cdot C_{r + 1} + C_{2} \cdot C_{r + 2} + . . . . + C_{n - r} \cdot C_{n} =^{2 n} C_{(n + r)}

and hence deduce that
i)

C_{0}^{2} + C_{1}^{2} + C_{2}^{2} + . . . . . . + C_{n}^{2} =^{2 n} C_{n}

ii)

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

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

(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + \dots + C_{n} x^{n},

then the value of

\sum \sum 0 \leq r < s \leq n {(C_{r} + C_{s})}_{r}^{2}

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Find the value of C0C3+C1C4+.....Cn−3Cn, when Cr=nCr

Find the value of $C_{0} C_{3} + C_{1} C_{4} + . . . . . C_{n - 3} C_{n}$ , when $C_{r} =^{n} C_{r}$