A group of 7 members having a majority of boys to be formed out of 6 boys and 4 girls. The number of ways the group can be formed is?

Majority means that there must be more boys than girls in the group.  The ways that can happen given the pools to draw from are

6 boys 1 girl

5 boys 2 girls

4 boys 3 girls

and that's it.

Let's look at the middle case, 5 boys and 2 girls.

The number of ways to choose 5 boys out of 6 is C(6, 5) = 6! / (5! 1!) = 6

The number of ways to choose 2 girls out of 4 is C(4, 2) = 4! / (2! 2!) = 6

So the number of combinations for the middle case is 6 * 6 = 36

Now do the other two cases, and add up the results.