The number of ways in which the letters of the word SUMPTUOS can be arranged so that the two U's do not come together is:

Options

A. a. 5040
B. b. 7560
C. c. 38920
D. d. none of (a),(b),(C )

Answer

To find the number of arrangements of the letters in the word "SUMPTUOS" such that the two U's do not come together, we can follow these steps:

### Step 1: Calculate the total arrangements of the letters in "SUMPTUOS".
The word "SUMPTUOS" has 8 letters, where S appears 2 times and U appears 2 times. The formula for the total arrangements of letters when there are repetitions is given by:

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

where \( n \) is the total number of letters, and \( p_1, p_2, \ldots \) are the frequencies of the repeated letters.

Here, \( n = 8 \) (total letters), \( p_1 = 2 \) (for S), and \( p_2 = 2 \) (for U).

So,

\[ \text{Total arrangements} = \frac{8!}{2! \times 2!} \]

### Step 2: Calculate the arrangements where the two U's are together.
To find the arrangements where the two U's are together, we can treat the two U's as a single entity or block. Thus, we have the following letters to arrange: (UU), S, S, M, P, T, O. This gives us a total of 7 entities (the block of U's counts as one entity).

Now, we can calculate the arrangements of these 7 entities, where S appears 2 times:

\[ \text{Arrangements with U's together} = \frac{7!}{2!} \]

### Step 3: Calculate the arrangements where the two U's do not come together.
To find the arrangements where the two U's do not come together, we subtract the arrangements where the U's are together from the total arrangements:

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

Substituting the values we calculated:

\[ \text{Arrangements with U's not together} = \frac{8!}{2! \times 2!} - \frac{7!}{2!} \]

### Step 4: Simplify the expression.
Calculating \( 8! \) and \( 7! \):

\[ 8! = 40320 \quad \text{and} \quad 7! = 5040 \]

Now substituting these values:

\[ \text{Total arrangements} = \frac{40320}{2 \times 2} = \frac{40320}{4} = 10080 \]
\[ \text{Arrangements with U's together} = \frac{5040}{2} = 2520 \]

Now we can find the arrangements where the U's do not come together:

\[ \text{Arrangements with U's not together} = 10080 - 2520 = 7560 \]

### Final Answer:
The number of ways in which the letters of the word "SUMPTUOS" can be arranged such that the two U's do not come together is **7560**.

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