If for 1- m- n-fm-n - C 0 - C 1 - C 2 - -1 m-1 C m-1- find f9-5-

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Sum of Coefficients of All Terms

If for 1≤ m...

Question

If for $1 \leq m \leq n, f (m, n)$ $= C_{0} - C_{1} + C_{2} - . . . . {(- 1)}^{m - 1} C_{m - 1}$ , find $f (9, 5)$ .

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

n \in N,

2 \leq m \leq n,

C_{0} - C_{1} + C_{2} - \dots + [(- 1)^{m - 1}] C_{m - 1}

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

r \geq 0

(\begin{matrix} m r \end{matrix}) C_{0} + (\begin{matrix} m r - 1 \end{matrix}) C_{1} + . . . . . . (\begin{matrix} m 0 \end{matrix}) C_{r} = (\begin{matrix} m + n r \end{matrix})

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

(1 + x)^{n} .

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

If $^{n} C_{r} = C_{r}$ then $C_{0} + (C_{0} + C_{1}) + (C_{0} + C_{1} + C_{2}) + . . . . + (C_{0} + C_{1} + . . . . . + C_{n})$ =

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

C_{0}, C_{1}, C_{2}

n

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