How many words can be made from the word IMPORTANT in which both T do not come together?

Options

A. a. 141120
B. b. 112244
C. c. 113113
D. d. 888222

Answer

To solve the problem of how many words can be formed from the word "IMPORTANT" such that both T's do not come together, we can follow these steps:

### Step 1: Calculate the Total Arrangements of the Word "IMPORTANT"
The word "IMPORTANT" consists of 9 letters: I, M, P, O, R, T, A, N, T. The letter T appears twice, while all other letters appear once.

The formula for the total arrangements of letters, considering repetitions, is given by:

\[ \text{Total arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!} \]

Where:
- \( n \) is the total number of letters,
- \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeated letters.

For "IMPORTANT":
- Total letters, \( n = 9 \)
- The letter T appears 2 times.

Thus, the total arrangements are:

\[ \text{Total arrangements} = \frac{9!}{2!} \]

### Step 2: Calculate the Arrangements Where Both T's Are Together
To find the arrangements where both T's are together, we can treat the two T's as a single unit or block. This gives us the following letters to arrange: (TT), I, M, P, O, R, A, N.

Now, we have 8 units to arrange: (TT), I, M, P, O, R, A, N.

The total arrangements of these 8 units (where TT is treated as one unit) is:

\[ \text{Arrangements with T's together} = 8! \]

### Step 3: Calculate the Arrangements Where Both T's Do Not Come Together
To find the arrangements where both T's do not come together, we subtract the arrangements where both T's are together from the total arrangements:

\[ \text{Arrangements with T's not together} = \text{Total arrangements} - \text{Arrangements with T's together} \]

Substituting the values we calculated:

\[ \text{Arrangements with T's not together} = \frac{9!}{2!} - 8! \]

### Step 4: Simplify the Expression
Now, we can simplify the expression:

\[ \frac{9!}{2!} = \frac{9 \times 8!}{2} = \frac{9 \times 8!}{2} - 8! \]

Factoring out \( 8! \):

\[ = 8! \left( \frac{9}{2} - 1 \right) = 8! \left( \frac{9 - 2}{2} \right) = 8! \left( \frac{7}{2} \right) \]

### Step 5: Calculate the Final Answer
Now we calculate \( 8! \) and multiply it by \( \frac{7}{2} \):

\[ 8! = 40320 \]

Thus,

\[ \text{Arrangements with T's not together} = 40320 \times \frac{7}{2} = 40320 \times 3.5 = 141120 \]

### Final Answer
The total number of words that can be formed from the word "IMPORTANT" such that both T's do not come together is **141120**.

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