n C 0 - - C C 1- n C 2- n C 3 - - - - -A- 0B- 1C- 2nD- 2n-1

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Sum of product of binomial coefficients

n C 0 - - C C...

Question

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

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Solution

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Similar questions

$^{n} C_{0}$ + $^{n} C_{2}$ + $^{n} C_{4}$ ...................=

$^{n} C_{1}$ + $^{n} C_{3}$ + $^{n} C_{5}$ ..............=

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

^{2 n + 1} C_{0} +^{2 n + 1} C_{1} +^{2 n + 1} C_{2} + \dots +^{2 n + 1} C_{n} = 4^{11}

Which of the following is/are true?

(1) $(1 + i)^{n}$ = $2^{\frac{n}{2}}$ $(s i n \frac{n π}{4} + i c o s \frac{n π}{4})$

If $(1 + i)^{n}$ = $^{n} C_{0}$ + $^{n} C_{1}$ i - $^{n} C_{2}$ - $^{n} C_{3}$ i + $^{n} C_{4}$ ...............

(2) $^{n} C_{0}$ - $^{n} C_{2}$ + $^{n} C_{4}$ - $^{n} C_{6}$ ................ = $2^{\frac{n}{2}}$ $s i n \frac{n π}{4}$

(3) $^{n} C_{1}$ - $^{n} C_{3}$ + $^{n} C_{5}$ - $^{n} C_{7}$ ....................= $2^{\frac{n}{2}} c o s \frac{n π}{4}$

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