The sum S n - k -0 n -1 K -3 n C k- isA- -1n-3 n-1 Cn-1B- -1n-3 n-1 Cn-1C- None of these D- -1n-3 n-1 Cn

The correct option is D S_n=^3nC_0 ^3nC_1 + ^3nC_2 +..... + ( 1)^n. ^3nC_n But ^3nC_0 = ^3n 1C_0 ^3nC_1 = ^3n 1C_0 ^ 3n 1C_1 ^3nC_2 = ^3n 1C_1 +^3n 1 ...

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Byju's Answer

Standard XII

Mathematics

Combination

The sum S n =...

Question

The sum $S_{n} = \sum_{k = 0}^{n} (- 1)^{K} .^{3 n} C_{k},$ is

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None of these

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Solution

The correct option is B
$S_{n} =^{3 n} C_{0} -^{3 n} C_{1} +^{3 n} C_{2} + . . . . . + (- 1)^{n} .^{3 n} C_{n} B u t^{3 n} C_{0} =^{3 n - 1} C_{0} -^{3 n} C_{1} = -^{3 n - 1} C_{0} -^{3 n - 1} C_{1}^{3 n} C_{2} =^{3 n - 1} C_{1} +^{3 n - 1} C_{2} -^{3 n} C_{3} = -^{3 n - 1} C_{2} -^{3 n - 1} C_{3} . . . . . . . . . . . . . . . . . . . . . . . . . . . . (- 1)^{n} .^{3 n} C_{n} = (- 1)^{n} .^{3 n - 1} C_{1} + (- 1)^{n} .^{3 n - 1} C_{n}$
On adding we get $S_{n} = (- 1)^{n} .^{3 n - 1} C_{n}$

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Similar questions

Q. The sum

S_{n} = \sum_{k = 0}^{n} (- 1)^{K} .^{3 n} C_{k}, w h e r e n = 12, . . . . . .

Q. The value of

\frac{1}{2}^{n} C_{1} .^{n - 1} C_{1} + \frac{2}{3}^{n} C_{2} + \frac{3}{4}^{n} C_{c} .^{n - 1} C_{3} + \dots \dots

Let $f (n) = ∣ ∣ ∣ ∣ ∣ \begin{matrix} n & n + 1 & n + 2^{n} P_{n} & ^{n + 1} P_{n + 1} & ^{n + 1} P_{n + 2}^{n} C_{n} & ^{n + 1} C_{n + 1} & ^{n + 1} C_{n + 2} \end{matrix} ∣ ∣ ∣ ∣ ∣$
Then, f(n) is divisible by

Find the value of $^{n + 1} C_{1}$ - $^{n + 1} C_{2}$ + $^{n + 1} C_{3}$ .................. $(- 1)^{n + 1}$ $^{n + 1} C_{n + 1}$

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

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The sum Sn=∑nk=0(−1)K.3nCk, is

The correct option is B Sn=3nC0−3nC1+3nC2+.....+(−1)n.3nCnBut3nC0=3n−1C0−3nC1=−3n−1C0−3n−1C13nC2=3n−1C1+3n−1C2−3nC3=−3n−1C2−3n−1C3............................(−1)n.3nCn=(−1)n.3n−1C1+(−1)n.3n−1Cn On adding we get Sn=(−1)n.3n−1Cn

The sum $S_{n} = \sum_{k = 0}^{n} (- 1)^{K} .^{3 n} C_{k},$ is