### Core Logic Letters in 'DISTRIBUTION': I(3), T(2), D, S, R, B, U, O, N. There are 9 distinct letters. ### Step 1: Case Analysis **Case 1: 3 alike, 1 distinct** Selection: Choose the letter 'I' (^1C_1$^1C_1$) and 1 from the remaining 8 distinct letters (^8C_1$^8C_1$). Arrangement: ^8C_1 times frac4!3! = 8 times 4 = 32$^8C_1 \times \frac{4!}{3!} = 8 \times 4 = 32$. **Case 2: 2 alike of one kind, 2 alike of another kind** Since only 'I' and 'T' appear at least twice, we must choose both. Arrangement: ^2C_2 times frac4!2!2! = 1 times 6 = 6$^2C_2 \times \frac{4!}{2!2!} = 1 \times 6 = 6$. ### Step 2: Further Cases **Case 3: 2 alike, 2 distinct** Selection: Choose 1 from the 2 repeated sets (^2C_1$^2C_1$) and 2 from the remaining 8 distinct letters (^8C_2$^8C_2$). Arrangement: ^2C_1 times ^8C_2 times frac4!2! = 2 times 28 times 12 = 672$^2C_1 \times ^8C_2 \times \frac{4!}{2!} = 2 \times 28 \times 12 = 672$. **Case 4: All 4 distinct** Selection: Choose 4 from the 9 distinct letters (^9C_4$^9C_4$). Arrangement: ^9C_4 times 4! = 126 times 24 = 3024$^9C_4 \times 4! = 126 \times 24 = 3024$. ### Step 3: Total Words Total = 3024 + 672 + 6 + 32 = 3734$3024 + 672 + 6 + 32 = 3734$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Maths: Permutations and Combinations