All possible products are formed from the numbers `1, 2, 3, 4, ..., 200` by selecting any two without repetition. The number of products out of the total obtained which are multiples of 5 is :

Options

A. 5040
B. 7180
C. 8150
D. none of these

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the total number of products formed by selecting any two numbers from 1 to 200.

The total number of ways to select 2 numbers from 200 is given by the combination formula \( nCk \), where \( n \) is the total number of items to choose from, and \( k \) is the number of items to choose.

\[ \text{Total products} = \binom{200}{2} = \frac{200 \times 199}{2} = 19900 \]

### Step 2: Identify the numbers that are multiples of 5.

The multiples of 5 from 1 to 200 are: 5, 10, 15, ..., 200. This forms an arithmetic sequence where:
- The first term \( a = 5 \)
- The common difference \( d = 5 \)
- The last term \( l = 200 \)

To find the number of terms \( n \) in this sequence, we can use the formula for the nth term of an arithmetic sequence:

\[ l = a + (n-1)d \implies 200 = 5 + (n-1) \cdot 5 \]
\[ 200 - 5 = (n-1) \cdot 5 \implies 195 = (n-1) \cdot 5 \implies n-1 = 39 \implies n = 40 \]

Thus, there are 40 multiples of 5 between 1 and 200.

### Step 3: Calculate the number of numbers that are not multiples of 5.

The total numbers from 1 to 200 is 200. Therefore, the numbers that are not multiples of 5 are:

\[ \text{Numbers not multiples of 5} = 200 - 40 = 160 \]

### Step 4: Calculate the number of products formed by selecting two numbers that are not multiples of 5.

The number of ways to select 2 numbers from the 160 numbers that are not multiples of 5 is:

\[ \text{Products not multiples of 5} = \binom{160}{2} = \frac{160 \times 159}{2} = 12720 \]

### Step 5: Calculate the number of products that are multiples of 5.

To find the number of products that are multiples of 5, we subtract the number of products that are not multiples of 5 from the total number of products:

\[ \text{Products that are multiples of 5} = \text{Total products} - \text{Products not multiples of 5} \]
\[ = 19900 - 12720 = 7180 \]

### Final Answer:

The number of products that are multiples of 5 is **7180**.

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