Question

How many different strings can be made by reordering the letters of the word SUCCESS?

• A. \(512\)
• B. \(56\)
• C. \(420\)
• D. \(24\)

Hint:

For arrangements with repeated letters: \[ \text{Number of arrangements} = \frac{n!}{p_1!p_2!\cdots}. \] Always divide by the factorials of the frequencies of repeated letters.

Answer 1

The Correct Option is C

Concept: When all objects are distinct, the number of arrangements of \(n\) objects is \[ n!. \] However, if some objects are repeated, identical arrangements are counted multiple times. Therefore, for repeated objects, the number of distinct arrangements is \[ \frac{n!}{p_1!p_2!p_3!\cdots}, \] where \(p_1,p_2,p_3,\ldots\) are the frequencies of repeated objects. This formula is known as the permutation formula with repetition.

Step 1: Count the total number of letters.
The word \[ SUCCESS \] contains the letters \[ S,\;U,\;C,\;C,\;E,\;S,\;S. \] Therefore the total number of letters is \[ 7. \]

Step 2: Determine the repeated letters.
The letter \(S\) appears \[ 3 \] times. The letter \(C\) appears \[ 2 \] times. The letters \(U\) and \(E\) each appear once. Thus the repetitions are \[ S^3,\qquad C^2. \]

Step 3: Apply the formula for permutations with repetition.
The number of distinct arrangements is \[ \frac{7!}{3!\,2!}. \] Substituting the factorial values, \[ = \frac{5040}{(6)(2)}. \] \[ = \frac{5040}{12}. \] \[ = 420. \]

Step 4: Verify the result.
If all letters were distinct, the number of arrangements would be \[ 7!=5040. \] Since three \(S\)'s and two \(C\)'s are identical, we divide by \[ 3!\times2! \] to remove duplicate counting. Hence the answer remains \[ 420. \]

Step 5: Write the final answer.
Therefore the number of different strings that can be formed is \[ \boxed{420}. \] Hence the correct option is \[ \boxed{(C)\;420}. \]