Probability of Sum 22 or More with Four Dice

Total outcomes = $6^4 = 1296$. Possible ways to get sum $\ge 22$:

• Sum = 24: (6,6,6,6) $\rightarrow 1$ way
• Sum = 23: (6,6,6,5) $\rightarrow \frac{4!}{3!} = 4$ ways
• Sum = 22: (6,6,6,4) $\rightarrow 4$ ways, (6,6,5,5) $\rightarrow \frac{4!}{2!2!} = 6$ ways Total favorable outcomes = $1 + 4 + 4 + 6 = 15$. Probability = $\frac{15}{1296} = \frac{5}{432}$.