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

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

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

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

Commented by Tawa11 last updated on 13/Jul/22

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