How many different combinations of quarters, nickels, and dimes can be used to make $0.55?
+26367 $$\small{\text{Quater $ = 25 \qquad $Dime $ = 10 \qquad $ Nickel $= 5
$}} \\
\small{\text{In \$0.55 max. 2 Quaters, max. 5 Dimes and max. 11 Nickels }} \\\\
\left(\sum\limits_{i=0}^{2} x^{(\textcolor[rgb]{0,0,1}{25}*i}) \right) \times \left (\sum\limits_{i=0}^{5} x^{(\textcolor[rgb]{0,0,1}{10}*i)} \right) \times \left(\sum\limits_{i=0}^{11} x^{(\textcolor[rgb]{0,0,1}{5}*i)} \right) \\\\
\small{\text{$=(1+x^{25}+x^{50})$}} \times
\small{\text{$(1+x^{10}+x^{20}+x^{30}+x^{40}+x^{50})$}} \\ \times
\small{\text{$(1+x^{5}+x^{10}+x^{15}+x^{20}+x^{25}+x^{30}+x^{35}+x^{40}+x^{45}+x^{50}+x^{55})$}}\\
\small{\text{$
=(x^{155})+(x^{150})+2*x^{145}+2*x^{140}+3*x^{135}+4*x^{130} $}} \\
\small{\text{$+5*x^{125}+6*x^{120}+7*x^{115}+8*x^{110}+10*x^{105}+11*x^{100} $}} \\
\small{\text{$+11*x^{95}+12*x^{90}+12*x^{85}+13*x^{80}+13*x^{75}+12*x^{70} $}} \\
\small{\text{$+12*x^{65}+11*x^{60}+\textcolor[rgb]{1,0,0}{11*x^{55}}+10*x^{50}+8*x^{45}+7*x^{40} $}} \\
\small{\text{$+6*x^{35}+5*x^{30}+4*x^{25}+3*x^{20}+2*x^{15}+2*x^{10}+(x^5)+1 $}} \\$$
The coefficient from $$\small{\text{$\textcolor[rgb]{1,0,0}{x^{55}}$}}$$ is $$\small{\text{$\textcolor[rgb]{1,0,0}{ 11 }$}}$$. So there are 11 possibilities.
+128460 Let's see.......
2 quarters 1 nickel
1 quarter 3 dimes
1 quarter 2 dimes 2 nickels
1 quarter 1 dime 4 nickels
1 quarter 6 nickels
5 dimes 1 nickel
4 dimes 3 nickels
3 dimes 5 nickels
2 dimes 7 nickels
1 dime 9 nickels
11 nickels
I think that's it......did I leave anything out ???
+26367
How many different combinations of quarters, nickels, and dimes can be used to make $0.55 ?
$$\small{\text{
$
\begin{array}{rccrlclcl}
\hline
\\
1. & \$0.55 &=& & && 3\; Dimes &+& 1\; Quater \\
2. & \$0.55 &=& 1& Nickel && &+& 2\; Quaters \\
3. & \$0.55 &=& 1& Nickel &+& 5\; Dimes \\
4. & \$0.55 &=& 2& Nickels &+& 2\; Dimes &+& 1\; Quater \\
5. & \$0.55 &=& 3& Nickels &+& 4\; Dimes \\
6. & \$0.55 &=& 4& Nickels &+& 1\; Dime &+& 1\; Quater\\
7. & \$0.55 &=& 5& Nickels &+& 3\; Dimes \\
8. & \$0.55 &=& 6& Nickels && &+& 1\; Quater \\
9. & \$0.55 &=& 7& Nickels &+& 2\; Dimes \\
10. & \$0.55 &=& 9& Nickels &+& 1\; Dime \\
11. & \$0.55 &=& 11& Nickels \\
\\
\hline
\end{array}
$}}$$
+26367 $$\small{\text{Quater $ = 25 \qquad $Dime $ = 10 \qquad $ Nickel $= 5
$}} \\
\small{\text{In \$0.55 max. 2 Quaters, max. 5 Dimes and max. 11 Nickels }} \\\\
\left(\sum\limits_{i=0}^{2} x^{(\textcolor[rgb]{0,0,1}{25}*i}) \right) \times \left (\sum\limits_{i=0}^{5} x^{(\textcolor[rgb]{0,0,1}{10}*i)} \right) \times \left(\sum\limits_{i=0}^{11} x^{(\textcolor[rgb]{0,0,1}{5}*i)} \right) \\\\
\small{\text{$=(1+x^{25}+x^{50})$}} \times
\small{\text{$(1+x^{10}+x^{20}+x^{30}+x^{40}+x^{50})$}} \\ \times
\small{\text{$(1+x^{5}+x^{10}+x^{15}+x^{20}+x^{25}+x^{30}+x^{35}+x^{40}+x^{45}+x^{50}+x^{55})$}}\\
\small{\text{$
=(x^{155})+(x^{150})+2*x^{145}+2*x^{140}+3*x^{135}+4*x^{130} $}} \\
\small{\text{$+5*x^{125}+6*x^{120}+7*x^{115}+8*x^{110}+10*x^{105}+11*x^{100} $}} \\
\small{\text{$+11*x^{95}+12*x^{90}+12*x^{85}+13*x^{80}+13*x^{75}+12*x^{70} $}} \\
\small{\text{$+12*x^{65}+11*x^{60}+\textcolor[rgb]{1,0,0}{11*x^{55}}+10*x^{50}+8*x^{45}+7*x^{40} $}} \\
\small{\text{$+6*x^{35}+5*x^{30}+4*x^{25}+3*x^{20}+2*x^{15}+2*x^{10}+(x^5)+1 $}} \\$$
The coefficient from $$\small{\text{$\textcolor[rgb]{1,0,0}{x^{55}}$}}$$ is $$\small{\text{$\textcolor[rgb]{1,0,0}{ 11 }$}}$$. So there are 11 possibilities.
