Probability of rolling $14$ with $3$ dice vs. $5$ dice

Question

Here's a problem from my probability book:

The chance of throwing $14$ is greater with three dice than with five as $54$ to $49$.

Unfortunately, I got that the chance of both is the same! So where did I go wrong? Let me show my work:

Let's first do three dice:

• $662$: $3$ possibilities
• $653$: $6$ possibilities
• $644$: $3$ possibilities
• $554$: $3$ possibilities

Adding them up, we have$${{3 + 6 + 3 + 3}\over{6^3}} = {5\over{72}}$$

Let's next do five dice:

• $65111$: $20$ possibilities
• $64211$: $60$ possibilities
• $63311$: $30$ possibilities
• $63221$: $60$ possibilities
• $62222$: $5$ possibilities
• $55211$: $30$ possibilities
• $54311$: $60$ possibilities
• $54221$: $60$ possibilities
• $53321$: $60$ possibilities
• $54222$: $20$ possibilities
• $44411$: $10$ possibilities
• $44321$: $60$ possibilities
• $44222$: $10$ possibilities
• $43331$: $20$ possibilities
• $43322$: $30$ possibilities
• $33332$: $5$ possibilities

Adding them up, we have$${{540}\over{6^5}} = {5\over{72}},$$which shouldn't be the case since the two probabilities shouldn't be equal. So where did I overcount in the case of $5$ dice? Or is there a mistake in the problem statement?

Answer 1

There must be a mistake in the problem statement. I've checked with a few lines of python: there are indeed $15$ ways to score exactly $14$ with three dice and $540$ ways with five. I haven't been able to find a plausible way to correct it: changing $14$ to another number, changing "exactly" to "at most", or "less than", all fail to give the claimed ratio.

Answer 2

Lets the generating function of each dice be $$(x + x^2 + x^3 +x^4 +x^5 +x^6)$$.

Then for the sum of $14$ for $3$ dice , $$[x^{14}](x + x^2 + x^3 +x^4 +x^5 +x^6)^3$$ , it is $15$ , check it :https://www.wolframalpha.com/input/?i=expanded+form+of+%28x+%2B+x%5E2+%2B+x%5E3+%2Bx%5E4+%2Bx%5E5+%2Bx%5E6%29%5E3

Then for the sum of $14$ for $5$ dice, $$[x^{14}](x + x^2 + x^3 +x^4 +x^5 +x^6)^5$$ , it is $540$ ,check it : https://www.wolframalpha.com/input/?i=expanded+form+of+%28x+%2B+x%5E2+%2B+x%5E3+%2Bx%5E4+%2Bx%5E5+%2Bx%5E6%29%5E5

There may be a mistake in question