If n is a positive integer and `x_1,x_2,x_3,.......,x_k in C` then
`(x_1 + x_2 +x_3 + .............+ x_k)^n = sum (n!)/((alpha_1!)(alpha_2!)(alpha_3!)...........(alpha_k!))x_1^(alpha_1) x_2^(alpha_2) x_3^(alpha_3)............x_k^(alpha_k)`
where `alpha_1 , alpha_2 , alpha_3 , .................. , alpha_k` are all non-negative integers such that
`alpha_1 + alpha_2 + alpha_3 + ........+ alpha_k= n`
`text(Note)`
`(i)` The coefficient of `x_1^(alpha_1) x_2^(alpha_2) x_3^(alpha_3)............x_k^(alpha_k)` in the expansion of `(x_1 + x_2 +x_3 + .............+ x_k)^n` is `sum (n!)/((alpha_1!)(alpha_2!)(alpha_3!)`
`(ii)` `(a+b+c)^n = sum (n!)/((alpha!)(beta!)(gamma!)) a^alphab^betac^gamma` such that `alpha + beta + gamma = n`
`(iii)` `(a+b+c+d)^n = sum (n!)/((alpha!)(beta!)(gamma!)(delta!))a^alphab^betac^gammad^delta` such that `alpha + beta + gamma + delta = n`
`text(Number of Distinct or Dissimilar Terms in the Multinomial Expansion)`
Statement The number of distinct or dissimilar terms in the multinomial
expansion of `(x_1 + x_2 + x_3 + .............+ x_k)^n` is `text( )^(n+k-1)C_(k-1) `
`text(Proof)`
We have
`(x_1 + x_2 +x_3 + .............+ x_k)^n = sum (n!)/((alpha_1!)(alpha_2!)(alpha_3!) x_1^(alpha_1) x_2^(alpha_2) x_3^(alpha_3)............x_k^(alpha_k)`
where `alpha_1 ,alpha_2 ,alpha_3 ,................,alpha_k` are non-negative integers such that
`alpha_1 + alpha_2 + alpha_3 + ............ + alpha_k = n....................(i) `
Here, the number of terms in the expansion of `(x_1 + x_2 +x_3 + .............+ x_k)^n `
= The number of non-negative integral solutions of the Eq. (i)
`= text( )^(n+k-1)C_(k-1) `
`text(Greatest Coefficient in Multinomial Expansion)`
The greatest coefficient in the expansion of `(x_1 + x_2 +x_3 + .............+ x_k)^n ` is `(n!)/((q!)^(k-r)((q+1)!)^r),` where q is the quotient and r is the remainder when n is divided by k i.e.,
`text(Coefficient of)` `x^r` `text(in Multinomial Expansion)`
If n is a positive integer and `a_1,a_2,a_3,........,a_k in C` then coefficient of `x^r` in the
expansion of `(a_1 + a_2x + a_3x^2 + ..+ a_k x^(k-1) )^n ,` is
`sum (n!)/((alpha_1!)(alpha_2!)(alpha_3!) x_1^(alpha_1) b_2^(alpha_2) c_3^(alpha_3)............a_k^(alpha_k)`
where `alpha_1 ,alpha_2 ,alpha_3 ,................,alpha_k` are non-negative integers such that
`alpha_1 + alpha_2 + alpha_3 + ............ + alpha_k = n`
and `alpha_2 + 2alpha_3 + 3 alpha_4 + ......... + (k-1) alpha_k=r`
If n is a positive integer and `x_1,x_2,x_3,.......,x_k in C` then
`(x_1 + x_2 +x_3 + .............+ x_k)^n = sum (n!)/((alpha_1!)(alpha_2!)(alpha_3!)...........(alpha_k!))x_1^(alpha_1) x_2^(alpha_2) x_3^(alpha_3)............x_k^(alpha_k)`
where `alpha_1 , alpha_2 , alpha_3 , .................. , alpha_k` are all non-negative integers such that
`alpha_1 + alpha_2 + alpha_3 + ........+ alpha_k= n`
`text(Note)`
`(i)` The coefficient of `x_1^(alpha_1) x_2^(alpha_2) x_3^(alpha_3)............x_k^(alpha_k)` in the expansion of `(x_1 + x_2 +x_3 + .............+ x_k)^n` is `sum (n!)/((alpha_1!)(alpha_2!)(alpha_3!)`
`(ii)` `(a+b+c)^n = sum (n!)/((alpha!)(beta!)(gamma!)) a^alphab^betac^gamma` such that `alpha + beta + gamma = n`
`(iii)` `(a+b+c+d)^n = sum (n!)/((alpha!)(beta!)(gamma!)(delta!))a^alphab^betac^gammad^delta` such that `alpha + beta + gamma + delta = n`
`text(Number of Distinct or Dissimilar Terms in the Multinomial Expansion)`
Statement The number of distinct or dissimilar terms in the multinomial
expansion of `(x_1 + x_2 + x_3 + .............+ x_k)^n` is `text( )^(n+k-1)C_(k-1) `
`text(Proof)`
We have
`(x_1 + x_2 +x_3 + .............+ x_k)^n = sum (n!)/((alpha_1!)(alpha_2!)(alpha_3!) x_1^(alpha_1) x_2^(alpha_2) x_3^(alpha_3)............x_k^(alpha_k)`
where `alpha_1 ,alpha_2 ,alpha_3 ,................,alpha_k` are non-negative integers such that
`alpha_1 + alpha_2 + alpha_3 + ............ + alpha_k = n....................(i) `
Here, the number of terms in the expansion of `(x_1 + x_2 +x_3 + .............+ x_k)^n `
= The number of non-negative integral solutions of the Eq. (i)
`= text( )^(n+k-1)C_(k-1) `
`text(Greatest Coefficient in Multinomial Expansion)`
The greatest coefficient in the expansion of `(x_1 + x_2 +x_3 + .............+ x_k)^n ` is `(n!)/((q!)^(k-r)((q+1)!)^r),` where q is the quotient and r is the remainder when n is divided by k i.e.,
`text(Coefficient of)` `x^r` `text(in Multinomial Expansion)`
If n is a positive integer and `a_1,a_2,a_3,........,a_k in C` then coefficient of `x^r` in the
expansion of `(a_1 + a_2x + a_3x^2 + ..+ a_k x^(k-1) )^n ,` is
`sum (n!)/((alpha_1!)(alpha_2!)(alpha_3!) x_1^(alpha_1) b_2^(alpha_2) c_3^(alpha_3)............a_k^(alpha_k)`
where `alpha_1 ,alpha_2 ,alpha_3 ,................,alpha_k` are non-negative integers such that
`alpha_1 + alpha_2 + alpha_3 + ............ + alpha_k = n`
and `alpha_2 + 2alpha_3 + 3 alpha_4 + ......... + (k-1) alpha_k=r`