Solution:

To find the number of different batches of candidates that can be chosen out of ten candidates to fill six seats of articled clerks in a Chartered Accountant Firm, we need to use the concept of combinations.

Combinations refer to the number of ways of selecting a group of objects from a larger set, without considering the order of objects. The formula to calculate the number of combinations is:

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

where n is the total number of objects, r is the number of objects to be selected, n! represents the factorial of n (i.e., the product of all positive integers up to n), and r! * (n - r)! represents the factorial of (n - r) multiplied by the factorial of r.

Using this formula, we can calculate the number of different batches of candidates as follows:

Step 1: Determine the values of n and r.

Here, n = 10 (total number of candidates) and r = 6 (number of seats to be filled).

Step 2: Substitute the values in the formula and simplify.

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

nC6 = 10! / 6! * (10 - 6)!

nC6 = 10! / 6! * 4!

nC6 = (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / [(6 * 5 * 4 * 3 * 2 * 1) * (4 * 3 * 2 * 1)]

nC6 = (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1)

nC6 = 210

Therefore, the number of different batches of candidates that can be chosen out of ten candidates to fill six seats of articled clerks in a Chartered Accountant Firm is 210.

Conclusion:

Hence, the solution to the given problem is to choose 210 different batches of candidates out of ten candidates to fill six seats of articled clerks in a Chartered Accountant Firm.

10c4=10.9.8.7.6!÷4.3.2.1.6!
= 210 Ans