Combinations of percentages.

Question

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%.

Answer 1

Following this we can find what you want from the generating function of partitions without repetition based on $k$ numbers for $k\in \{7,8,17,21,23,24\}$. Then

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

Now, from the previous expression, we must count the coefficients of the exponents that are equal or bigger to $51$, that is if we set

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

then $f(1)$ is what you want.