Write - r -0 m n - r C r in the simplified form-

∑_r = 0^m^n+rC_r = ^nC_0 + ^n+1C_1 + ^n+2C_2 + ^n+3C_3 + ... + ^n+mC_m = n!/0! n! + (n+1)!/1! n! + (n+2)!/2! n! + (n+3)!/3! n! + ... + (n+m)!/n! m! = (m!) (n!) ...

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Byju's Answer

Standard XII

Mathematics

Sum of Binomial Coefficients of Odd Numbered Terms

Write ∑ r =0 ...

Question

Write $\sum_{r = 0}^{m}^{n + r} C_{r}$ in the simplified form.

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Solution

$\sum_{r = 0}^{m}^{n + r} C_{r}$
$=^{n} C_{0} +^{n + 1} C_{1} +^{n + 2} C_{2} +^{n + 3} C_{3} + . . . +^{n + m} C_{m}$
$= \frac{n!}{0! n!} + \frac{(n + 1)!}{1! n!} + \frac{(n + 2)!}{2! n!} + \frac{(n + 3)!}{3! n!} + . . . + \frac{(n + m)!}{n! m!} = \frac{(m!) (n!) + (n + 1)! m! + (n + 2)! m! + . . . + (n + m)! m!}{n! m!} = \frac{(n + m + 1)!}{m! (n + 1)!} =^{n + m + 1} C_{n + 1}$

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Similar questions

Q. Write

\sum_{r = 0}^{m} {}^{n + r}C_{r}

in the simplified form.

Q. The value of

n \sum r = 0

^{n + r} C_{r}

is equal to

Q. The sum of

n \sum r = 0 (- 1)^{r} \frac{^{n} C_{r}}{^{r + 2} C_{r}}

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

If n is an odd natural number , then $n \sum r = 0 \frac{(- 1)^{r}}{^{r} C_{r}}$ equals :

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Write ∑mr=0n+rCr in the simplified form.

∑mr=0n+rCr =nC0+n+1C1+n+2C2+n+3C3+...+n+mCm =n!0!n!+(n+1)!1!n!+(n+2)!2!n!+(n+3)!3!n!+...+(n+m)!n!m!=(m!)(n!)+(n+1)!m!+(n+2)!m!+...+(n+m)!m!n!m!=(n+m+1)!m!(n+1)!=n+m+1Cn+1

Write $\sum_{r = 0}^{m}^{n + r} C_{r}$ in the simplified form.