The is the number of ways of getting coefficient. If you look at the above expanded product the only way to get coefficient is when from the first two binomals we choose and from one of the last four binomials we choose . There are 4 ways of doing that so .
The is the number of ways of getting coefficint. If you look at the above expanded product you can get if you choose in first binomial and let others be or if you choose first one to be second binomal to be and all others to be . This already gives us two ways. But another way is to choose from the first two binomials and then choose two 's from the last four. Thus, there are ways of doing that so .
The term can be got by choosing from first binomial and pairing it with one of the 's from the last four binomials and so there are ways already. Similar argument involving in the second binomial gives us another ways. And finally we can choose from the first two binomials and then 3 's from the last four binomials. Therefore, .