# Probability That the Sum of 7 Dice is 26 Given: - 7 fair six-faced dice are rolled. - Each die shows a number from 1 to 6. - We need the probability that the sum is **26**. --- ## 1. **Total Number of Outcomes** Each die has 6 faces, so: \[ \text{Total outcomes} = 6^7 = 279,\!936 \] --- ## 2. **Counting Favorable Outcomes** We must count the number of ordered 7-tuples \((x_1, x_2, ..., x_7)\) such that: - \(x_1 + x_2 + ... + x_7 = 26\) - \(1 \leq x_i \leq 6\) for each \(i\) ### **Transform the Variables** Let \(y_i = x_i - 1\). Then \( \leq y_i \leq 5\), and: \[ x_1 + x_2 + ... + x_7 = 26 \implies y_1 + y_2 + ... + y_7 = 26 - 7 = 19 \] Now, find non-negative integer solutions to: \[ y_1 + y_2 + ... + y_7 = 19,\quad \leq y_i \leq 5 \] ### **Use Inclusion-Exclusion Principle** Let \(A_i\) be the set where \(y_i \geq 6\). The number of solutions without restriction is: \[ \text{Unrestricted:} \quad \binom{19 + 7 - 1}{7-1} = \binom{25}{6} = 177,\!100 \] We must subtract cases where one or more \(y_i > 5\): #### **For one variable \((y_i \geq 6)\):** Set \(y_i' = y_i - 6\) (\(y_i' \geq \)), so: \[ y_1' + y_2 + \cdots + y_7 = 19 - 6 = 13 \] Number of solutions for one \(y_i \geq 6\): \[ \binom{13 + 7 - 1}{7-1} = \binom{19}{6} = 27,\!132 \] There are 7 choices for which variable exceeds 5: \[ 7 \times 27,\!132 = 189,\!924 \] #### **For two variables \((y_i, y_j \geq 6)\):** Each such pair, set \(y_i', y_j' \geq \): \[ y_1' + y_2' + y_3 + ... + y_7 = 19 - 12 = 7 \] Number of solutions: \[ \binom{7 + 7 - 1}{7-1} = \binom{13}{6} = 1,\!716 \] Number of pairs: \(\binom{7}{2} = 21\): \[ 21 \times 1,\!716 = 36,\!036 \] #### **For three variables \((y_i, y_j, y_k \geq 6)\):** \[ y_1' + y_2' + y_3' + y_4 + ... + y_7 = 19 - 18 = 1 \] Number of solutions: \[ \binom{1 + 7 - 1}{7-1} = \binom{7}{6} = 7 \] Number of triples: \(\binom{7}{3} = 35\): \[ 35 \times 7 = 245 \] #### **For four or more variables:** If 4 dice have at least 6, the minimum sum is \(6 \times 4 = 24 > 19\), which is not possible. So, only up to three variables can exceed 5. --- ### **Apply Inclusion-Exclusion Principle (IEP)** \[ N = \binom{25}{6} - \binom{7}{1}\binom{19}{6} + \binom{7}{2}\binom{13}{6} - \binom{7}{3}\binom{7}{6} \] Plug in the numbers: \[ N = 177,\!100 - 189,\!924 + 36,\!036 - 245 \] \[ N = (177,\!100 + 36,\!036) - (189,\!924 + 245) \] \[ N = 213,\!136 - 190,\!169 = 22,\!967 \] --- ## 3. **Calculate the Probability** \[ P = \frac{22,\!967}{279,\!936} \] \[ P \approx .082 \] --- ## 4. **Final Answer** **The probability that the sum of the faces is 26 is:** \[ \boxed{\dfrac{22,\!967}{279,\!936}} \] or approximately \[ \boxed{.082} \]