Number of ways to select 3 chairs out of 7:

To solve this problem, we can use the concept of combinations. The number of ways to select 3 chairs out of 7 is given by the formula for combinations, which is denoted as nCr, where n is the total number of items and r is the number of items to be selected. The formula for combinations is:

nCr = n! / (r!(n-r)!)

where "!" denotes the factorial operation.

In this case, we have 7 chairs and we want to select 3 of them. Plugging these values into the formula, we get:

7C3 = 7! / (3!(7-3)!)
= 7! / (3!4!)

Now, let's simplify the expression:

7! = 7 x 6 x 5 x 4 x 3 x 2 x 1
3! = 3 x 2 x 1
4! = 4 x 3 x 2 x 1

Canceling out the common factors, we have:

7C3 = (7 x 6 x 5 x 4 x 3 x 2 x 1) / (3 x 2 x 1 x 4 x 3 x 2 x 1)
= (7 x 6 x 5) / (3 x 2 x 1)
= 35

Therefore, there are 35 ways to select 3 chairs out of 7.

Number of ways to arrange the 3 persons on the selected chairs:

Once we have selected the 3 chairs, we need to arrange the 3 persons on them. Since the order of arrangement matters, we can use the concept of permutations.

The number of ways to arrange 3 persons on 3 chairs is given by the formula for permutations, which is denoted as nPr, where n is the total number of items and r is the number of items to be arranged. The formula for permutations is:

nPr = n! / (n-r)!

In this case, we have 3 persons and 3 chairs. Plugging these values into the formula, we get:

3P3 = 3! / (3-3)!
= 3! / 0!

Since 0! is defined as 1, the expression simplifies to:

3P3 = 3!

Calculating the factorial of 3:

3! = 3 x 2 x 1
= 6

Therefore, there are 6 ways to arrange the 3 persons on the selected chairs.

Total number of ways:

To find the total number of ways, we need to multiply the number of ways to select the chairs (35) by the number of ways to arrange the persons (6):

Total number of ways = 35 x 6
= 210

Hence, the correct answer is option 'B' (210).