how many 4 letter words can be formed from letters of ‘MAHARAJA’

The word "MAHARAJA" consists of 8 letters, but some of these letters are repeated. Specifically, the letter "A" appears 4 times, the letter "R" appears 2 times, and the letters "M", "H", and "J" each appear once.

To find how many distinct 4-letter words can be formed, we need to consider the different combinations of letters and how they can be arranged. We break it down by cases based on how many repeated letters are in the word:

### Case 1: All 4 letters are different.
Here, we choose 4 distinct letters from {M, A, H, R, J}. Since A appears multiple times, we only need to consider one A. So, the distinct letters are {M, A, H, R, J}, a total of 5 distinct letters. 

The number of ways to choose 4 letters from 5 is \( \binom{5}{4} = 5 \), and for each choice of 4 letters, there are \(4!\) ways to arrange them.

\[
\text{Total for Case 1} = \binom{5}{4} \times 4! = 5 \times 24 = 120
\]

### Case 2: 1 letter appears twice, and 2 other letters are different.
Here, we choose one of the letters that repeats twice: {A, R}. There are 2 such letters. After choosing the repeated letter, we choose 2 distinct letters from the remaining distinct letters {M, A, H, R, J} (excluding the repeated letter).

The number of ways to choose the repeated letter is 2, and the number of ways to choose 2 different letters from the remaining 4 distinct letters is \( \binom{4}{2} = 6 \). These 3 letters (including the one repeated) can be arranged in \( \frac{4!}{2!} = 12 \) ways (since one letter is repeated twice).

\[
\text{Total for Case 2} = 2 \times 6 \times 12 = 144
\]

### Case 3: 1 letter appears three times, and 1 other letter is different.
Here, we choose one of the letters that can appear three times. Only the letter "A" can be chosen in this case. After choosing "A" three times, we choose 1 other letter from the distinct letters {M, H, R, J}.

The number of ways to choose the other letter is 4, and these 2 distinct letters (with one repeated three times) can be arranged in \( \frac{4!}{3!} = 4 \) ways (since one letter is repeated three times).

\[
\text{Total for Case 3} = 4 \times 4 = 16
\]

### Case 4: 2 letters appear twice.
Here, we choose 2 letters that each appear twice. The letters that appear more than once are {A, R}. So, we choose both "A" and "R". These 2 distinct letters can be arranged in \( \frac{4!}{2!2!} = 6 \) ways (since both letters are repeated twice).

\[
\text{Total for Case 4} = 6
\]

### Final Total
Now, adding up all the cases:

\[
120 \, (\text{Case 1}) + 144 \, (\text{Case 2}) + 16 \, (\text{Case 3}) + 6 \, (\text{Case 4}) = 286
\]

Thus, **286** distinct 4-letter words can be formed from the letters of "MAHARAJA".

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### Related questions:
1. How many 3-letter words can be formed from the letters of "MAHARAJA"?
2. How many 5-letter words can be formed from the letters of "MAHARAJA"?
3. What is the probability of randomly forming a word with no repeated letters from "MAHARAJA"?
4. How many 4-letter words can be formed if the letters need to be in alphabetical order?
5. How many distinct ways can we arrange all the letters of "MAHARAJA"?

### Tip:
When dealing with repeated letters, always consider using permutations with repetition formulas.

How many 4-letter words can be formed from letters of ‘MAHARAJA’?

Discover how to calculate the number of distinct 4-letter words that can be formed using the letters of 'MAHARAJA'. This solution covers combinatorics and permutations with repetition, suitable for high school students in Grades 10-12.

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