To solve this problem, we need to consider the different combinations of selecting 5 people from a group of 4 gentlemen and 6 ladies. We need to find the number of ways in which the committee can be formed such that the gentlemen are in the majority.

Step 1: Determine the number of ways to select 5 people from a group of 10.
The number of ways to select 5 people from a group of 10 can be calculated using the combination formula:

C(n, r) = n! / (r!(n-r)!)

In this case, n = 10 (total number of people) and r = 5 (number of people to be selected). Plugging in the values:

C(10, 5) = 10! / (5!(10-5)!)

C(10, 5) = 10! / (5!5!)

C(10, 5) = (10*9*8*7*6*5!) / (5!*5!)

C(10, 5) = (10*9*8*7*6) / (5*4*3*2*1)

C(10, 5) = 252

So, there are 252 ways to select 5 people from a group of 10.

Step 2: Determine the number of ways to have gentlemen in the majority.
To have gentlemen in the majority, there must be at least 3 gentlemen and at most 2 ladies in the committee. We can calculate this by considering different cases:

Case 1: 3 gentlemen and 2 ladies
The number of ways to select 3 gentlemen from a group of 4 can be calculated using the combination formula:

C(4, 3) = 4! / (3!(4-3)!)

C(4, 3) = 4! / (3!1!)

C(4, 3) = (4*3*2*1) / (3*2*1*1)

C(4, 3) = 4

The number of ways to select 2 ladies from a group of 6 can be calculated using the combination formula:

C(6, 2) = 6! / (2!(6-2)!)

C(6, 2) = 6! / (2!4!)

C(6, 2) = (6*5*4!) / (2!4!)

C(6, 2) = (6*5) / (2*1)

C(6, 2) = 15

So, there are 4 ways to select 3 gentlemen and 15 ways to select 2 ladies.

Case 2: 4 gentlemen and 1 lady
The number of ways to select 4 gentlemen from a group of 4 can be calculated using the combination formula:

C(4, 4) = 4! / (4!(4-4)!)

C(4, 4) = 4! / (4!0!)

C(4, 4) = (4*3*2*1) / (4*3*2*1*1)

C(4, 4) = 1

The number of ways to select 1 lady from a group of 6 can be calculated using the combination formula:

C