JEE Main & Advanced Mathematics Binomial Theorem and Mathematical Induction Multinomial Theorem (for Positive Integral Index)

If \[n\] is positive integer and \[{{a}_{1}},\,{{a}_{2}},\,{{a}_{3}},\,....{{a}_{n}}\in C\], then

\[{{({{a}_{1}}+{{a}_{2}}+{{a}_{3}}+...+{{a}_{m}})}^{n}}=\sum{\frac{n!}{{{n}_{1}}!\,{{n}_{2}}!{{n}_{3}}!...{{n}_{m}}!}a_{1}^{{{n}_{1}}}.a_{2}^{{{n}_{2}}}}...a_{m}^{{{n}_{m}}}\],

where \[{{n}_{1}},\,{{n}_{2}},\,{{n}_{3}},.....{{n}_{m}}\] are all non-negative integers subject to the condition, \[{{n}_{1}}+{{n}_{2}}+{{n}_{3}}+.....{{n}_{m}}=n\].

(1) The coefficient of \[a_{1}^{{{n}_{1}}}.a_{2}^{{{n}_{2}}}.....a_{m}^{{{n}_{m}}}\] in the expansion of \[{{({{a}_{1}}+{{a}_{2}}+{{a}_{3}}+....{{a}_{m}})}^{n}}\] is  \[\frac{n!}{{{n}_{1}}!{{n}_{2}}!{{n}_{3}}!....{{n}_{m}}!}\]

(2) The greatest coefficient in the expansion of

\[{{({{a}_{1}}+{{a}_{2}}+{{a}_{3}}+....{{a}_{m}})}^{n}}\] is \[\frac{n!}{{{(q!)}^{m-r}}{{[(q+1)!]}^{r}}}\]

where \[q\] is the quotient and \[r\] is the remainder when \[n\]is divided by \[m\].

(3) If \[n\] is \[+ve\] integer and \[{{a}_{1}},\,{{a}_{2}},.....{{a}_{m}}\in C\] \[a_{1}^{{{n}_{1}}}\,.\,a_{2}^{{{n}_{2}}}\,.........a_{m}^{{{n}_{m}}}\] then coefficient of \[{{x}^{r}}\] in the expansion of \[{{({{a}_{1}}+{{a}_{2}}x+.....{{a}_{m}}{{x}^{m-1}})}^{n}}\] is \[\sum{\frac{n!}{{{n}_{1}}!{{n}_{2}}!{{n}_{3}}!.....{{n}_{m}}!}}\]         

where \[{{n}_{1}},\,{{n}_{2}},.....{{n}_{m}}\] are all non-negative integers subject to the condition:

\[{{n}_{1}}+{{n}_{2}}+.....{{n}_{m}}=n\] and \[{{n}_{2}}+2{{n}_{3}}+3{{n}_{4}}+....+(m-1){{n}_{m}}=r\]

(4) The number of distinct or dissimilar terms in the multinomial expansion \[{{({{a}_{1}}+{{a}_{2}}+{{a}_{3}}+....{{a}_{m}})}^{n}}\] is \[^{n+m-1}{{C}_{m-1}}\].