Question

From 6 gentlemen and 4 ladies a committee of 5 is to be formed. Number of ways can this be done if

(a) The committee is to include at least one lady.
(b) there is no restriction about its formation.
A. 240,255
B. 242,256
C. 246,252
D. none of these

Answer 1

To determine the number of ways to form a committee of 5 members from 6 gentlemen and 4 ladies, we use combinatorial formulas, subtracting the ways to form a committee without any ladies for part (a), and without any restrictions for part (b). The correct answer is 246 ways to form a committee with at least one lady and 252 ways without any restrictions. So the correct answer is option C.

Step-by-step explanation:

The problem in question is a combinatorial problem concerning the formation of a committee with certain restrictions. Specifically, it's a problem about counting the number of ways to select a committee from a larger group.

Part (a): Including at least one lady

To ensure at least one lady is on the committee, we can subtract the number of committees with no ladies from the total number of possible committees. The total ways to form a committee of 5 from 10 people (6 gentlemen and 4 ladies) is given by the combination formula C(n, k) = n! / (k!(n-k)!), which in this case is C(10, 5). The number of committees with no ladies is C(6, 5) since we are choosing all 5 members from the 6 gentlemen. So, the number of committees with at least one lady is C(10, 5) - C(6, 5).

Part (b): No restriction on formation

If there are no restrictions, then any group of 5 people is a valid committee. This is simply C(10, 5).

Calculating these combinations, we find:

Total number of committees without restriction = 252

Thus, the correct answer is C. 246,252.