Seven different letters are given. Then the number of ways in which , words of 5 letters can be formed such that atleast one of the letters is repeated is

Options

A. P(7,5)
B. 14287
C. `7^(5)`
D. `5^(7)`

Answer

To solve the problem of finding the number of ways to form 5-letter words from 7 different letters such that at least one letter is repeated, we can follow these steps:

### Step 1: Calculate the total number of 5-letter words without any restrictions.
Since we have 7 different letters and each of the 5 positions in the word can be filled by any of these 7 letters, the total number of unrestricted 5-letter words can be calculated as follows:

\[ \text{Total 5-letter words} = 7^5 \]

Calculating \(7^5\):

\[ 7^5 = 7 \times 7 \times 7 \times 7 \times 7 = 16807 \]

### Step 2: Calculate the number of 5-letter words with all different letters.
If we want to form a 5-letter word using 7 different letters, and we want all letters to be different, we can choose the letters for each position as follows:

- The first position can be filled in 7 ways (any of the 7 letters).
- The second position can be filled in 6 ways (one letter has already been used).
- The third position can be filled in 5 ways.
- The fourth position can be filled in 4 ways.
- The fifth position can be filled in 3 ways.

Thus, the total number of 5-letter words with all different letters is:

\[ \text{Total different 5-letter words} = 7 \times 6 \times 5 \times 4 \times 3 \]

Calculating this:

\[ 7 \times 6 = 42 \]
\[ 42 \times 5 = 210 \]
\[ 210 \times 4 = 840 \]
\[ 840 \times 3 = 2520 \]

### Step 3: Calculate the number of 5-letter words with at least one letter repeated.
To find the number of 5-letter words with at least one letter repeated, we subtract the number of 5-letter words with all different letters from the total number of unrestricted 5-letter words:

\[ \text{Number of words with at least one letter repeated} = \text{Total 5-letter words} - \text{Total different 5-letter words} \]

Substituting the values we calculated:

\[ \text{Number of words with at least one letter repeated} = 16807 - 2520 = 14287 \]

### Final Answer:
The number of ways in which 5-letter words can be formed such that at least one letter is repeated is **14287**.

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