The number of ways in which we can select 5 letters of the word INTERNATIONAL is equal to

Options

A. 200
B. 220
C. 242
D. 256

Answer

To solve the problem of selecting 5 letters from the word "INTERNATIONAL," we need to consider the repetitions of letters in the word. The letters in "INTERNATIONAL" are as follows:

- I: 1
- N: 2
- T: 2
- E: 1
- R: 1
- A: 2
- O: 1
- L: 1

Now, let's break down the solution step by step:

### Step 1: Count the Unique Letters
The unique letters in "INTERNATIONAL" are I, N, T, E, R, A, O, L. This gives us a total of 8 unique letters.

### Step 2: Case Analysis
We will analyze different cases based on the repetition of letters.

#### Case 1: All 5 letters are different
We can choose 5 letters from 8 unique letters. The number of ways to choose 5 different letters from 8 is given by the combination formula \( \binom{n}{r} \):

\[ \text{Ways} = \binom{8}{5} = \frac{8!}{5!(8-5)!} = \frac{8!}{5! \cdot 3!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \]

#### Case 2: 3 letters are the same, and 2 are different
The only letter that can be chosen 3 times is 'N', 'T', or 'A'. We will consider 'N' for this case.

- Choose 2 different letters from the remaining 7 letters (I, T, E, R, A, O, L):

\[ \text{Ways} = 1 \cdot \binom{7}{2} = 1 \cdot \frac{7!}{2!(7-2)!} = 1 \cdot \frac{7 \times 6}{2 \times 1} = 21 \]

#### Case 3: 2 letters are the same, and 3 are different
We can choose any letter that appears twice (N, T, or A) and then choose 3 different letters from the remaining 7 letters.

- Choose 1 letter to repeat (3 choices: N, T, A) and then choose 3 different letters from the remaining 7:

\[ \text{Ways} = 3 \cdot \binom{7}{3} = 3 \cdot \frac{7!}{3!(7-3)!} = 3 \cdot \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 3 \cdot 35 = 105 \]

#### Case 4: 2 letters are the same, and 2 letters are the same, and 1 is different
We can choose 2 letters that appear twice (N, T, A) and then choose 1 different letter from the remaining 6 letters.

- Choose 2 letters to repeat (3 choices for pairs: NT, NA, TA) and then choose 1 different letter from the remaining 6:

\[ \text{Ways} = 3 \cdot \binom{6}{1} = 3 \cdot 6 = 18 \]

### Step 3: Total Ways
Now, we add all the cases together:

\[ \text{Total Ways} = 56 + 21 + 105 + 18 = 200 \]

### Final Answer
Thus, the total number of ways to select 5 letters from the word "INTERNATIONAL" is **200**.

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