The number of permutations of the alphabets of the word ''GOOGLE'' in which O's are together but G's are separated, is

Options

A. 24
B. 48
C. 72
D. 36

Answer

To solve the problem of finding the number of permutations of the letters in the word "GOOGLE" such that the two O's are together and the two G's are separated, we can follow these steps:

### Step-by-Step Solution:

1. **Consider the O's as a Single Unit:**
Since the two O's must be together, we can treat them as a single unit or packet. Therefore, we can represent the word "GOOGLE" as:
- OO (the packet of O's)
- G
- G
- L
- E

This gives us the units: {OO, G, G, L, E}.

2. **Count the Total Units:**
Now we have a total of 5 units to arrange: {OO, G, G, L, E}.

3. **Calculate the Arrangements of the Units:**
The number of ways to arrange these 5 units, where the G's are identical, is given by the formula for permutations of multiset:
\[ \text{Number of arrangements} = \frac{n!}{p_1! \cdot p_2! \cdot \ldots} \]
Here, \( n = 5 \) (total units), and we have 2 G's which are identical. Thus:
\[ \text{Arrangements} = \frac{5!}{2!} = \frac{120}{2} = 60 \]

4. **Identify the Gaps for G's:**
Next, we need to ensure that the two G's are separated. When we arrange the units, we can visualize the arrangement as having gaps where G's can be placed.

For example, if we arrange the other units (OO, L, E), we can visualize them as:
- _ OO _ L _ E _
This creates 4 gaps (before OO, between OO and L, between L and E, and after E).

5. **Choose Gaps for G's:**
We need to select 2 out of these 4 gaps to place the G's. The number of ways to choose 2 gaps from 4 is given by the combination formula:
\[ \text{Ways to choose gaps} = \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

6. **Calculate Total Valid Arrangements:**
Finally, to find the total number of arrangements where the O's are together and the G's are separated, we multiply the number of arrangements of the units by the number of ways to choose the gaps:
\[ \text{Total arrangements} = \text{Arrangements of units} \times \text{Ways to choose gaps} = 60 \times 6 = 360 \]

### Final Answer:
The total number of permutations of the letters in "GOOGLE" such that the O's are together and the G's are separated is **360**.