# Combinations of percentages.

How would I calculate all possible combinations of given percentages so that none of the combinations is less than 51%? For example one such combination of

• 24%
• 23%
• 21%
• 17%
• 8%
• 7%

would be 23% + 24% + 7% = 54%.

• with our without repetition? For example $7 /{%} + 7 /{%} + 8 /{%}$ ? – Slepecky Mamut Oct 6 '14 at 14:16
• @SlepeckyMamut Without repetition, hence percentages. – ikaruss Oct 6 '14 at 14:18
• @Paul 32% is less than 51% and therefore of no interest. – ikaruss Oct 6 '14 at 14:19

## 1 Answer

Following this we can see this when we create the generating function of partitions without repetition based on k numbers where $k\in \{7,8,17,21,23,24\}$. So

$$SQ=(1+x^7)(1+x^8)(1+x^{17})(1+x^{21})(1+x^{23})(1+x^{24})=\\ =x^{100}+x^{93}+x^{92}+x^{85}+x ^{83}+x^{79}+x^{77}+2 x^{76}+x^{75}+x^{72}+x^{71} +x^{70}+2 x^{69}+2 x^{68}+x^{64}+2 x^{62}+x^{61}+x^{60}+x^{59} +x^{56}+2 x^{55}+x^{54}+2 x^{53}+2 x^{52}+x^{51}+x^{49}+2 x^{48}+2 x^{47}+x^{46}+2 x^{45}+x^{44}+x^{41}+x^{40} +x^{39}+2 x^{38}+x^{36}+2 x^{32}+2 x^{31}+x^{30}+x^{29}+x^{28} +x^{25}+2 x^{24}+x^{23}+x^{21}+x^{17} +x^{15}+x^8+x^7+1$$

From this polynomial we must count the coefficients of exponents that are equal or bigger to 51

$$f(x)=x^{100}+x^{93}+x^{92}+x^{85}+x ^{83}+x^{79}+x^{77}+2 x^{76}+x^{75}+x^{72}+x^{71} +x^{70}+2 x^{69}+2 x^{68}+x^{64}+2 x^{62}+x^{61}+x^{60}+x^{59} +x^{56}+2 x^{55}+x^{54}+2 x^{53}+2 x^{52}+x^{51}$$

Now f(1) is your number ;). May exist a more simple way but Im not sure/dont know now.