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Question Number 173371 by sujeet924 last updated on 10/Jul/22

The number of distinct terms in the expassion of (x_1 +x_2  +x_3  +.......+ x_n  )^(4 ) is  (a)

$${The}\:{number}\:{of}\:{distinct}\:{terms}\:{in}\:{the}\:{expassion}\:{of}\:\left({x}_{\mathrm{1}} +{x}_{\mathrm{2}} \:+{x}_{\mathrm{3}} \:+.......+\:{x}_{{n}} \:\right)^{\mathrm{4}\:} {is} \\ $$$$\left({a}\right)\: \\ $$

Commented by mr W last updated on 10/Jul/22

in expansion of (x_1 +x_2 +x_3 +...+x_n )^m   the general term is x_1 ^k_1  x_2 ^k_2  x_3 ^k_3  ...x_n ^k_n    with 0≤k_i ≤m and k_1 +k_2 +k_3 +...+k_n =m  there are C_m ^(n+m−1)  possibilities. i.e.  the number of distinc terms is  C_m ^(n+m−1) .   in case of m=4, it′s C_4 ^(n+3) .

$${in}\:{expansion}\:{of}\:\left({x}_{\mathrm{1}} +{x}_{\mathrm{2}} +{x}_{\mathrm{3}} +...+{x}_{{n}} \right)^{{m}} \\ $$$${the}\:{general}\:{term}\:{is}\:{x}_{\mathrm{1}} ^{{k}_{\mathrm{1}} } {x}_{\mathrm{2}} ^{{k}_{\mathrm{2}} } {x}_{\mathrm{3}} ^{{k}_{\mathrm{3}} } ...{x}_{{n}} ^{{k}_{{n}} } \\ $$$${with}\:\mathrm{0}\leqslant{k}_{{i}} \leqslant{m}\:{and}\:{k}_{\mathrm{1}} +{k}_{\mathrm{2}} +{k}_{\mathrm{3}} +...+{k}_{{n}} ={m} \\ $$$${there}\:{are}\:{C}_{{m}} ^{{n}+{m}−\mathrm{1}} \:{possibilities}.\:{i}.{e}. \\ $$$${the}\:{number}\:{of}\:{distinc}\:{terms}\:{is} \\ $$$${C}_{{m}} ^{{n}+{m}−\mathrm{1}} .\: \\ $$$${in}\:{case}\:{of}\:{m}=\mathrm{4},\:{it}'{s}\:{C}_{\mathrm{4}} ^{{n}+\mathrm{3}} . \\ $$

Commented by Tawa11 last updated on 13/Jul/22

Great sir

$$\mathrm{Great}\:\mathrm{sir} \\ $$

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