If c 0- c 1- c 2- - c 15 are the Binomial co-efficients in the expansion of 1- x 15- then the value of C 1- c 0-2 c 2- c 1-3 C 3- c 2-15 c 15- c 14 isA- 1240B- 120- 124D- 140

The correct option is B 120c_1/c_0+2 c_2/c_1+3 c_3/c_2+⋯+ 15 c_15/c_14 = 15+2 ( 15 1/2 )+3 ( 15 2/3 )+....+15 ( 15 14/15 ) [∵^nC_r^nC_r 1=n r+1r] =15+14+13+⋯ ...

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Byju's Answer

Standard XII

Mathematics

Expansions to Remove Indeterminate Form

If c 0, c 1, ...

Question

If $c_{0}, c_{1}, c_{2}, \dots c_{15}$ are the Binomial co-efficients in the expansion of $(1 + x)^{15}$ , then the value of $\frac{c_{1}}{c_{0}} + 2 \frac{c_{2}}{c_{1}} + 3 \frac{c_{3}}{c_{2}} + \dots + 15 \frac{c_{15}}{c_{14}}$ is

1240

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120

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124

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140

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Solution

The correct option is B $120$
$\frac{c_{1}}{c_{0}} + 2 \frac{c_{2}}{c_{1}} + 3 \frac{c_{3}}{c_{2}} + \dots + 15 \frac{c_{15}}{c_{14}} = 15 + 2 (\frac{15 - 1}{2}) + 3 (\frac{15 - 2}{3}) + . . . . + 15 (\frac{15 - 14}{15}) [∵ \frac{^{n} C_{r}}{^{n} C_{r - 1}} = \frac{n - r + 1}{r}] = 15 + 14 + 13 + \dots + 1 = \frac{15 \times 16}{2} = 120$

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Similar questions

Q. If

C_{0}, C_{1}, C_{2}, . . . . . . C_{15}

are the binomial coefficients in the expansion of

(1 + x)^{15}

, prove that

\frac{C_{1}}{C_{0}} + 2 \frac{C_{2}}{C_{1}} + 3 \frac{C_{3}}{C_{2}} + . . . . . . . . . . . + 15 \frac{C_{15}}{C_{14}} = 120

Q. If

C_{0}, C_{1}, C_{2}, \dots, C_{15}

are binomial coefficients in

{(1 + x)}^{15}

, then

\frac{C_{1}}{C_{0}} + 2 \frac{C_{2}}{C_{1}} + 3 \frac{C_{3}}{C_{2}} + \dots + 15 \frac{C_{15}}{C_{14}}

is equal to

\frac{C_{1}}{C_{0}} + 2 \frac{C_{2}}{C_{1}} + 3 \frac{C_{3}}{C_{2}} + . . . . + 15 \frac{C_{15}}{C_{14}}

is equal to

Q. If

C_{0}, C_{1}, C_{2} . . . ., C_{n}

denote the binomial coefficients in the expansion of

{(1 + x)}^{n}

, then

\frac{C 1}{C 0} + 2 \frac{C 2}{C 1} + + 3 \frac{C 3}{C 2} + . . . . . + n \frac{C n}{C n - 1}

equals

Q. If

C_{0}, C_{1}, C_{2}, . . . C_{n}

are the binomial coefficients in the expansion of

(1 + x)^{n}

then prove that:

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

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If c0,c1,c2,⋯c15 are the Binomial co-efficients in the expansion of (1+x)15 , then the value of c1c0+2c2c1+3c3c2+⋯+15c15c14 is

The correct option is B 120c1c0+2c2c1+3c3c2+⋯+15c15c14=15+2(15−12)+3(15−23)+....+15(15−1415)[∵nCrnCr−1=n−r+1r]=15+14+13+⋯+1=15×162=120

If $c_{0}, c_{1}, c_{2}, \dots c_{15}$ are the Binomial co-efficients in the expansion of $(1 + x)^{15}$ , then the value of $\frac{c_{1}}{c_{0}} + 2 \frac{c_{2}}{c_{1}} + 3 \frac{c_{3}}{c_{2}} + \dots + 15 \frac{c_{15}}{c_{14}}$ is