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7.Binomial Theorem
hard
$(1 + x)\,{(1 - x)^n}$ ના વિસ્તરણમાં ${x^n}$ નો સહગુણક મેળવો.
A
${( - 1)^{n - 1}}n$
B
${( - 1)^n}(1 - n)$
C
${( - 1)^{n - 1}}{(n - 1)^2}$
D
$(n - 1)$
(AIEEE-2004)
Solution
(b) Coefficient of ${x^n}$ in expansion of $(1 + x)$ ${(1 – x)^n}$ie.,
coefficient of ${x^n}$ in expansion of ${(1 – x)^n} + $ coefficient of ${x^{n – 1}}$ in expansion of ${(1 – x)^n}$
Now, ${( – 1)^n}{\,^n}{C_n} + {( – 1)^{n – 1}}\,{\,^n}{C_{n – 1}}$
$\therefore {( – 1)^n}\,{[^n}{C_n} – {\,^n}{C_{n – 1}}]$
= ${( – 1)^n}\,[1 – n]$.
Standard 11
Mathematics
