The number of ways in which 20 letters `a_(1),a_(2),a_(3),.............,a_(10),b_(1),b_(2),b_(3),.........,b_(10)` be arranged in a line so that suffixes of the letters 'a' and also those of 'b' are respectively in ascending orders of magnitude is………

Options

A. `(20!)/(10!)`
B. `(20!)/((10!)^(2))`
C. `2^(20)`
D. `20!-10!.10!`

Explanation

To solve the problem of arranging the letters \( a_1, a_2, \ldots, a_{10}, b_1, b_2, \ldots, b_{10} \) such that the suffixes of the letters 'a' and 'b' are in ascending order, we can follow these steps:

### Step 1: Understand the Arrangement Requirement
We need to arrange 20 letters (10 'a's and 10 'b's) in such a way that:
- The 'a's appear in the order \( a_1, a_2, \ldots, a_{10} \)
- The 'b's appear in the order \( b_1, b_2, \ldots, b_{10} \)

### Step 2: Choose Positions for 'a's
We can choose 10 positions out of the 20 available positions for the 'a's. The number of ways to choose 10 positions from 20 is given by the combination formula \( \binom{n}{r} \):
\[ \text{Number of ways to choose positions for 'a's} = \binom{20}{10} \]

### Step 3: Fill in the 'a's
Once we have chosen the positions for the 'a's, there is only one way to fill these positions with \( a_1, a_2, \ldots, a_{10} \) since they must be in ascending order.

### Step 4: Fill in the 'b's
After placing the 'a's, the remaining 10 positions will automatically be for the 'b's. Similar to the 'a's, there is only one way to fill these positions with \( b_1, b_2, \ldots, b_{10} \) since they must also be in ascending order.

### Step 5: Calculate the Total Arrangements
Since the arrangements of 'a's and 'b's are independent, the total number of arrangements is simply the number of ways to choose the positions for 'a's:
\[ \text{Total arrangements} = \binom{20}{10} \]

### Step 6: Compute the Value of \( \binom{20}{10} \)
Using the formula for combinations:
\[ \binom{20}{10} = \frac{20!}{10! \times 10!} \]

### Final Result
Thus, the total number of ways to arrange the letters is:
\[ \binom{20}{10} = \frac{20!}{10! \times 10!} \]