+94 (a) We have three standard 6-sided dice, colored red, yellow, and green. In how many ways can we roll them to get a sum of 9? The dice are colored so that a red 2, yellow 3, and green 4 is a different roll from red 3, yellow 4, and green 2.
(b) We have 10 standard 6-sided dice, all different colors. In how many ways can we roll them to get a sum of 20?
Please give an explanation.
+26367 (a)
We have three standard 6-sided dice, colored red, yellow, and green.
In how many ways can we roll them to get a sum of 9?
The dice are colored so that a red 2, yellow 3, and green 4 is a different roll from red 3, yellow 4, and green 2.
\(\begin{array}{|c|ccc|c|} \hline & \text{red} & \text{yellow} & \text{green} & \text{sum} \\ \hline 1 &1 &2 &6 &9 \\ 2 &1 &3 &5 &9 \\ 3 &1 &4 &4 &9 \\ 4 &1 &5 &3 &9 \\ 5 &1 &6 &2 &9 \\ 6 & 2&1 &6 &9 \\ 7 & 2&2 &5 &9 \\ 8 & 2&3 &4 &9 \\ 9 & 2&4 &3 &9 \\ 10 &2 &5 &2 &9 \\ 11 &2 &6 &1 &9 \\ 12 & 3&1 &5 &9 \\ 13 & 3&2 &4 &9 \\ 14 & 3&3 &3 &9 \\ 15 & 3&4 &2 &9 \\ 16 & 3&5 &1 &9 \\ 17 & 4&1 &4 &9 \\ 18 & 4&2 &3 &9 \\ 19 & 4&3 &2 &9 \\ 20 & 4&4 &1 &9 \\ 21 & 5&1 &3 &9 \\ 22 & 5&2 &2 &9 \\ 23 & 5&3 &1 &9 \\ 24 & 6&1 & 2&9 \\ 25 & 6&2 & 1&9 \\ \hline \end{array}\)
Remark:
The same table sorted by yellow
The same table sorted by green
We can roll them 25 ways to get a sum of 9
