The number of 5 digit numbers of the form xyzyz in which x < y is

Options

A. 350
B. 360
C. 380
D. 390

Answer

To solve the problem of finding the number of 5-digit numbers of the form xyzyz where x < y, we can follow these steps:

### Step 1: Understand the structure of the number
The number is in the form xyzyz, which means:
- The first digit is x.
- The second digit is y.
- The third digit is z.
- The fourth digit is y (same as the second).
- The fifth digit is z (same as the third).

### Step 2: Determine the constraints
Since x < y, we need to choose x and y such that this condition holds. The digits x and y can be any digit from 1 to 9 (since x cannot be 0 in a 5-digit number).

### Step 3: Choose x and y
We need to choose 2 different digits from the set {1, 2, 3, 4, 5, 6, 7, 8, 9}. The number of ways to choose 2 digits from 9 is given by the combination formula:
\[ \binom{9}{2} \]
This represents the number of ways to choose 2 digits out of 9.

### Step 4: Arranging x and y
For each pair of digits chosen, there are 2 arrangements: (x, y) and (y, x). However, since we only want the arrangement where x < y, we will only consider one of these arrangements.

### Step 5: Choose z
The digit z can be any digit from 0 to 9. This gives us 10 options for z.

### Step 6: Calculate the total combinations
The total number of valid combinations can be calculated as follows:
\[ \text{Total combinations} = \binom{9}{2} \times 10 \]
Since we are only considering the arrangement where x < y, we do not need to divide by 2.

### Step 7: Calculate \(\binom{9}{2}\)
Calculating \(\binom{9}{2}\):
\[ \binom{9}{2} = \frac{9!}{2!(9-2)!} = \frac{9 \times 8}{2 \times 1} = 36 \]

### Step 8: Final calculation
Now, substituting back into the total combinations:
\[ \text{Total combinations} = 36 \times 10 = 360 \]

### Conclusion
Thus, the total number of 5-digit numbers of the form xyzyz where x < y is **360**.