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7.Binomial Theorem
hard
$1 + (1 + x) + {(1 + x)^2} + ..... + {(1 + x)^n}$ ના વિસ્તરણમાં ${x^k}(0 \le k \le n)$ નો સહગુણક મેળવો.
A
$^{n + 1}{C_{k + 1}}$
B
$^n{C_k}$
C
$^n{C_{n - k - 1}}$
D
એકપણ નહિ.
Solution
(a)The expression being in $G. P.$
$E = 1 + (1 + x) + {(1 + x)^2} + …. + {(1 + x)^n}$
$\frac{{{{(1 + x)}^{n + 1}} – 1}}{{(1 + x) – 1}} = {x^{ – 1}}\{ {(1 + x)^{n + 1}} – 1\} $
$\therefore \,\,\,$The coefficient of $x^k $ in
$E =$ The coefficient of ${x^{k + 1}}$in $\{ {(1 + x)^{n + 1}} – 1\} $
$ = {\,^{n + 1}}{C_{k + 1}}$.
Standard 11
Mathematics
