Question

The total number of words (with or without meaning) that can be formed out of the letters of the word ‘DISTRIBUTION’ taken four at a time, is equal to _____

Answer 1

Correct Answer: 3734

The word DISTRIBUTION contains the letters: I, I, I, T, T, D, S, R, B, U, O, N. 

We calculate the number of distinct 4-letter words that can be formed by considering different cases of letter repetition:

1. Case 1: Three letters the same and one different (a, a, a, b) \[ \binom{4}{1} \times \frac{4!}{3!} = 32 \]
2. Case 2: Two letters each repeated twice (a, a, b, b) \[ \frac{4!}{2! \cdot 2!} = 6 \]
3. Case 3: Two identical letters and two distinct ones (a, b, c, c) \[ \binom{2}{1} \times \binom{2}{1} \times \frac{4!}{2!} = 672 \]
4. Case 4: All letters different (a, b, c, d) \[ \binom{4}{1} \times 4! = 3024 \]

Total number of possible words:

\[ \text{Total} = 3024 + 672 + 6 + 32 = 3734 \]

Answer 2

The letters in the word 'DISTRIBUTION' are: I, I, I, T, T, D, S, R, B, U, O, N.

Calculate the number of words formed using different combinations:

1. Number of words with selection (a, a, a, b):
\[ \binom{4}{1} \times \frac{4!}{3!} = 32 \]
2. Number of words with selection (a, a, b, b):
\[ \frac{4!}{2! \cdot 2!} = 6 \]
3. Number of words with selection (a, b, c, c):
\[ \binom{2}{1} \times \binom{2}{1} \times \frac{4!}{2!} = 672 \]
4. Number of words with selection (a, b, c, d):
\[ \binom{4}{1} \times 4! = 3024 \]

Total number of words:

\[ \text{Total} = 3024 + 672 + 6 + 32 = 3734 \]