In how many ways can the letters of the word "SUCCESS" be arranged if the two S's cannot be together?Choices:- 7!/2! - 6! 7!−6! 7!−2! 7!−5!

To solve this problem, we first need to find the total number of ways the letters can be arranged without any restrictions. Then we subtract the number of ways the letters can be arranged with the two S's together.

1. The word "SUCCESS" has 7 letters, but the letters S, C, and U are repeated. So, the total number of ways the letters can be arranged without any restrictions is 7!/(3!2!2!) = 420 ways.

2. To find the number of ways the letters can be arranged with the two S's together, we can treat the two S's as one letter. So, we now have 6 letters (the two S's, U, C, C, E). Again, the letters S, C, and U are repeated. So, the total number of ways these letters can be arranged is 6!/(2!2!) = 180 ways.

3. Finally, subtract the number of ways the letters can be arranged with the two S's together from the total number of ways the letters can be arranged without any restrictions. So, the answer is 420 - 180 = 240 ways.

Therefore, none of the given choices are correct.

To solve this problem, we first need to find the total number of ways the letters can be arranged without any restrictions. Then we subtract the number of ways the letters can be arranged with the two S's together.

1. Find the total number of ways the letters can be arranged without any restrictions:

The word "SUCCESS" has 7 letters, but the letters S, C, and U are repeated. So, we use the formula for permutations of a multiset: n! / (r1! * r2! * ... * rk!), where n is the total number of items, and r1, r2, ..., rk are the numbers of each type of repeated item.

In this case, n = 7 (the total number of letters), r1 = 3 (the number of S's), r2 = 2 (the number of C's), and r3 = 2 (the number of U's). So, the total number of ways is 7! / (3! * 2! * 2!) = 420.

2. Find the number of ways the letters can be arranged with the two S's together:

We can think of the two S's as a single item. So, we now have 6 items to arrange: the S's, U, U, C, C, E. Again, we have repeated items, so we use the formula for permutations of a multiset.

In this case, n = 6 (the total number of items), r1 = 2 (the number of U's), r2 = 2 (the number of C's). So, the number of ways is 6! / (2! * 2!) = 180.

3. Subtract the number of ways with the S's together from the total number of ways:

420 - 180 = 240.

So, there are 240 ways to arrange the letters of the word "SUCCESS" with the two S's not together.