Words of 5 letter are to be formed out of given 7 letters. If at least one letter is repeated, then the number of words so formed is

Options

A. `7^5`
B. `5^7`
C. 14287
D. `"^7 P_5`

Answer

To solve the problem of forming 5-letter words from 7 given letters with at least one letter repeated, we can follow these steps:

### Step 1: Calculate the total number of 5-letter words that can be formed with 7 letters (with repetition allowed).
Since we can repeat letters, each of the 5 positions in the word can be filled by any of the 7 letters. Therefore, the total number of 5-letter words is given by:

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

Calculating this:

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

### Step 2: Calculate the number of 5-letter words that can be formed without any letter being repeated.
To form a 5-letter word without repeating any letters, we need to choose 5 letters from the 7 available letters. The first letter can be any of the 7, the second letter can be any of the remaining 6, the third can be any of the remaining 5, the fourth can be any of the remaining 4, and the fifth can be any of the remaining 3. Therefore, the number of such arrangements is given by:

\[ \text{Non-repeated words} = 7 \times 6 \times 5 \times 4 \times 3 \]

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 words with at least one letter repeated, we subtract the number of non-repeated words from the total number of words:

\[ \text{Words with at least one repeated} = \text{Total words} - \text{Non-repeated words} \]

Substituting the values we calculated:

\[ \text{Words with at least one repeated} = 16807 - 2520 = 14287 \]

### Final Answer:
The number of 5-letter words that can be formed from 7 letters with at least one letter repeated is **14287**.

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