Gujarat BoardEnglish MediumSTD 11 CommerceStatisticsPermutations , combinations and binomial expansion2 Marks
Question
Write the characteristics of binomial expansion.
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Answer
Characteristics of binomial expansion :
The following characteristics are seen in the expansion of $(x+a)^n$ :
$(1)$ Total number of terms in the expansion is ( $n+1$ ).
$(2)$ The coefficients of these terms are ${ }^n C_0,{ }^n C_1{ }^n C_2, \ldots \ldots{ }^{ n } C_n$ respectively.
$(3)$ The sum of the coefficient of these terms : is $2^n$, i.e., ${ }^n C_0,+{ }^n C_1+{ }^n C_2 \ldots . .+{ }^n C_n=2^n$.
$(4)$ The first term is ${ }^n C_{o x}=x^n$ and last term is ${ }^n C_n a ^n= a ^n$.
$(5)$ In the successive terms. the power of ' $x$ ' goes on decreasing by one, while power of ' $a$ ' goes on increasing by one.
$(6)$ In any term the sum of the powers of $x$ and $a$ is $n$.
$(7)$ The coefficients of the terms equidistant from the middle term are equal.
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