In how many ways the word "PROPORTIONS" can be rearranged so that all O's come together?

To solve this problem, we need to find the number of ways to rearrange the letters in the word "PROPORTIONS" such that all the O's are together.

### Step-by-Step Solution

1. **Treat all O's as a single unit:**

   The word "PROPORTIONS" has 11 letters: P, R, O, P, O, R, T, I, O, N, S. There are 3 O's in the word. If we want all O's to come together, we can treat these 3 O's as a single "block." 

   So, instead of 11 separate letters, we now have 9 units to arrange: 
   - 1 "block of O's" (OOO)
   - P, R, P, R, T, I, N, S

2. **Count the number of arrangements of the 9 units:**

   Out of these 9 units, the letters P and R each appear twice. 

   The number of distinct arrangements of 9 units, accounting for the repetitions, is given by:

   \[
   \text{Number of arrangements} = \frac{9!}{2! \times 2!}
   \]

3. **Calculate the factorial values:**

   \[
   9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362880
   \]

   \[
   2! = 2 \times 1 = 2
   \]

   So,

   \[
   \text{Number of arrangements} = \frac{362880}{2 \times 2} = \frac{362880}{4} = 90720
   \]

### Final Answer

There are **90,720** ways to rearrange the letters in the word "PROPORTIONS" such that all O's come together.

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Would you like to know more details or have any questions? Here are 5 related questions to explore further:

1. How many ways can the word "PROPORTIONS" be rearranged without any restrictions?
2. What if two specific letters also need to stay together, along with the O's?
3. How would the number of arrangements change if there were 4 O's instead of 3?
4. How many ways can the word "PROPORTIONS" be rearranged so that no two O's are adjacent?
5. How many ways can the word "PROPORTIONS" be rearranged if all consonants need to be together?

**Tip:** When dealing with problems where certain elements must stay together, consider treating them as a single unit first.

In how many ways can the word 'PROPORTIONS' be rearranged so that all O's come together?

Discover the method to rearrange the letters in the word 'PROPORTIONS' such that all O's are together. This problem involves key combinatorial concepts like permutations and factorials, providing a detailed explanation suitable for students in Grades 10-12.

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