Previous in Permutation and Combination Next in Permutation and Combination
Question Number 22040 by Tinkutara last updated on 10/Oct/17
$$\mathrm{The}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{non}-\mathrm{similar} \\ $$$$\mathrm{triangles}\:\mathrm{which}\:\mathrm{can}\:\mathrm{be}\:\mathrm{formed}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{all}\:\mathrm{the}\:\mathrm{angles}\:\mathrm{of}\:\mathrm{the}\:\mathrm{triangle}\:\mathrm{are} \\ $$$$\mathrm{integers}\:\mathrm{is} \\ $$
Commented by mrW1 last updated on 01/Jan/18
$${the}\:{question}\:{is}\:{in}\:{how}\:{many}\:{ways} \\ $$$$\mathrm{180}\:{can}\:{be}\:{built}\:{as}\:{sum}\:{of}\:{three}\:{numbers} \\ $$$${which}\:{are}\:\geqslant\mathrm{1}.\:{this}\:{is}\:{the}\:{coefficient} \\ $$$${of}\:{x}^{\mathrm{180}} \:{term}\:{of}\:{following}\:{generating} \\ $$$${function}: \\ $$$$\frac{{x}^{\mathrm{3}} }{\left(\mathrm{1}−{x}\right)\left(\mathrm{1}−{x}^{\mathrm{2}} \right)\left(\mathrm{1}−{x}^{\mathrm{3}} \right)} \\ $$$${which}\:{is}\:\mathrm{2700}. \\ $$
Commented by mrW1 last updated on 01/Jan/18
Commented by Tinkutara last updated on 01/Jan/18
How do you find the expression for this generating function?
Commented by mrW1 last updated on 01/Jan/18
