To solve the problem of finding the number of permutations of the letters in the word "GOOGLE" where the two O's are together and the two G's are separated, we can follow these steps:
### Step 1: Treat the O's as a single unit
Since the two O's must be together, we can treat them as a single unit or block. Therefore, we can represent the word "GOOGLE" as:
- OO (the block of O's)
- G
- G
- L
- E
This gives us the units: {OO, G, G, L, E}.
### Step 2: Count the total units
Now, we have 5 units to arrange: {OO, G, G, L, E}.
### Step 3: Calculate the permutations of these units
The total number of arrangements of these 5 units, considering that the two G's are identical, is given by the formula for permutations of multiset:
\[
\text{Permutations} = \frac{n!}{p_1! \times p_2!}
\]
where \( n \) is the total number of items to arrange, and \( p_1, p_2, \ldots \) are the counts of identical items.
Here, we have:
- Total units \( n = 5 \) (OO, G, G, L, E)
- The G's are identical, so \( p_1 = 2! \)
Thus, the number of arrangements is:
\[
\text{Permutations} = \frac{5!}{2!} = \frac{120}{2} = 60
\]
### Step 4: Ensure G's are separated
Next, we need to ensure that the two G's are not together. To find this, we can use the gap method.
### Step 5: Create gaps for G's
When we arrange the units {OO, L, E}, we have:
- OO
- L
- E
This creates gaps:
- Before OO
- Between OO and L
- Between L and E
- After E
This gives us a total of 4 gaps:
1. _ (before OO)
2. _ (between OO and L)
3. _ (between L and E)
4. _ (after E)
### Step 6: Choose gaps for G's
We need to choose 2 out of these 4 gaps to place the G's, ensuring they are separated. The number of ways to choose 2 gaps from 4 is given by:
\[
\binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6
\]
### Step 7: Calculate total arrangements
Now, we multiply the number of arrangements of the units (where O's are together) by the number of ways to place the G's in separate gaps:
\[
\text{Total arrangements} = 60 \times 6 = 360
\]
### Conclusion
Thus, the total number of permutations of the letters in the word "GOOGLE" where the O's are together and the G's are separated is **360**.
