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Question Number 42224 by 33 last updated on 20/Aug/18

a + b + c = 180   a,b,c ∈ N  number of triplets possible  (a,b,c) for the above  equation are ?  ( the order of a,b,c  doesn′t  matter)

$${a}\:+\:{b}\:+\:{c}\:=\:\mathrm{180}\: \\ $$$${a},{b},{c}\:\in\:\mathbb{N} \\ $$$${number}\:{of}\:{triplets}\:{possible} \\ $$$$\left({a},{b},{c}\right)\:{for}\:{the}\:{above} \\ $$$${equation}\:{are}\:? \\ $$$$\left(\:{the}\:{order}\:{of}\:{a},{b},{c}\:\:{doesn}'{t}\right. \\ $$$$\left.{matter}\right) \\ $$

Commented by rahul 19 last updated on 20/Aug/18

15,931    =   (179_C_2  )     ?

$$\mathrm{15},\mathrm{931}\:\:\:\:=\:\:\:\left(\mathrm{179}_{\mathrm{C}_{\mathrm{2}} } \right)\:\:\:\:\:? \\ $$

Commented by 33 last updated on 20/Aug/18

answer is not known.  please  can you show the working.

$${answer}\:{is}\:{not}\:{known}. \\ $$$${please} \\ $$$${can}\:{you}\:{show}\:{the}\:{working}. \\ $$

Commented by rahul 19 last updated on 20/Aug/18

There is a direct formula for +ve  integers : n−1_C_(r−1)    For non. negative integers : n+r−1_C_(r−1)    They come from multinomeal theorem.  So in this case 179−1_C_(3−1)  .

$$\mathrm{There}\:\mathrm{is}\:\mathrm{a}\:\mathrm{direct}\:\mathrm{formula}\:\mathrm{for}\:+\mathrm{ve} \\ $$$$\mathrm{integers}\::\:\mathrm{n}−\mathrm{1}_{\mathrm{C}_{\mathrm{r}−\mathrm{1}} } \\ $$$$\mathrm{For}\:\mathrm{non}.\:\mathrm{negative}\:\mathrm{integers}\::\:\mathrm{n}+\mathrm{r}−\mathrm{1}_{\mathrm{C}_{\mathrm{r}−\mathrm{1}} } \\ $$$$\mathrm{They}\:\mathrm{come}\:\mathrm{from}\:\mathrm{multinomeal}\:\mathrm{theorem}. \\ $$$$\mathrm{So}\:\mathrm{in}\:\mathrm{this}\:\mathrm{case}\:\mathrm{179}−\mathrm{1}_{\mathrm{C}_{\mathrm{3}−\mathrm{1}} } . \\ $$

Commented by 33 last updated on 20/Aug/18

in case we dont know the  formula?

$${in}\:{case}\:{we}\:{dont}\:{know}\:{the} \\ $$$${formula}? \\ $$

Commented by Tinkutara last updated on 20/Aug/18

See Q. 22040

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