Using all the letters of the word CREATIVITY, how many distinct words or arrangements can be formed, considering that some letters are repeated and that the words may or may not have meaning?

Introduction / Context:
This question tests permutations with repeated letters. When forming arrangements from letters of a word that contains repeated characters, we must adjust the standard formula for permutations to avoid overcounting identical arrangements. This is a standard pattern in aptitude tests for permutations and combinations involving words like COMMITTEE or CREATIVITY.

Given Data / Assumptions:

• The word is CREATIVITY.
• The letters in order are C, R, E, A, T, I, V, I, T, Y.
• Total number of letters is 10.
• There are repeated letters: the letter I appears twice and the letter T appears twice.
• We count distinct arrangements, so arrangements that differ only by swapping identical letters are considered the same.

Concept / Approach:
For n objects where some objects are repeated, the number of distinct permutations is given by n! divided by the factorials of the frequencies of repeated objects. Here n is 10, and we have two I's and two T's. Hence we will compute 10! / (2! * 2!). This formula ensures that swapping identical letters does not produce artificially different arrangements in our count.

Step-by-Step Solution:
Step 1: Count the total number of letters: there are 10 letters in CREATIVITY.Step 2: Count the frequency of each letter: I appears twice, T appears twice, and the remaining letters C, R, E, A, V and Y each appear once.Step 3: Use the permutation formula for repeated letters: total arrangements = 10! / (2! * 2!).Step 4: Compute 10! = 3628800.Step 5: Compute 2! * 2! = 2 * 2 = 4.Step 6: Divide 3628800 by 4 to get 907200 distinct arrangements.
Verification / Alternative check:
If all letters were distinct, there would be 10! = 3628800 arrangements. However, because each pair of identical letters (I and T) leads to overcounting by a factor of 2 for each pair, we divide by 2 for the first pair and again by 2 for the second pair, which is equivalent to dividing by 4. The resulting count 907200 is reasonable and matches the standard formula for permutations with repeats.

Why Other Options Are Wrong:
The value 362880 corresponds to 9!, not 10! / (2! * 2!). The other numbers like 851250 or 751210 do not correspond to any correct application of factorial division for repeated letters. Only 907200 is consistent with the correct formula for this situation.

Common Pitfalls:
Students often forget to divide by factorials of repeated letters, which leads to overcounting. Others miscount the number of repeats, for example, missing that both I and T are repeated in CREATIVITY. Carefully listing frequencies before applying the formula helps avoid such mistakes in similar problems.

Final Answer:
The number of distinct arrangements that can be formed from the letters of CREATIVITY is 907200.