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7.Binomial Theorem
normal
જો $(1 + x)^m = C_0 + C_1x + C_2x^2 + C_3x^3 + . . . . . +C_mx^m$, જ્યાં $C_r ={}^m{C_r}$ અને $A = C_1C_3 + C_2C_4+ C_3C_5 + C_4C_6 + . . . . . .. + C_{m-2}C_m$, હોય તો નીચેનામાંથી ક્યુ ખોટું છે ?
A
$A \ge {}^{2m}{C_{m - 2}}$
B
$A < {}^{2m}{C_{m - 2}}$
C
$A = {}^{2m}{C_{m - 2}} - {}^m{C_2}$
D
$A < {C^2}_0 + {C^2}_1 + {C^2}_1 + .......{C^2}_m$
Solution
$\mathrm{A}=\mathrm{C}_{0} \mathrm{C}_{2}+\mathrm{C}_{1} \mathrm{C}_{3}+\mathrm{C}_{2} \mathrm{C}_{4}+\ldots+\mathrm{C}_{\mathrm{m}-2} \mathrm{C}_{\mathrm{m}}-\mathrm{C}_{0} \mathrm{C}_{2}$
$ = {\,^{2m}}{C_{m – 2}} – {\,^m}{C_2}$
and $C_0^2 + C_1^2 + \ldots \ldots + C_m^2{ = ^{2m}}{C_m}$
Standard 11
Mathematics