find the number of positive integral solutions of the equation xyz=210.

210 = 2 x 3  x 5 x 7. Since the problem asks for integers, we will first calculate for positive numbers and then assign signs to these numbers.

Your first instinct is to write down 2, 3, 5 and 7 in a row and place two partitions between them. The partitions will give you three groups of numbers. Something like this:

But you should quickly realize that in this way no matter where you place your partitions, the number 2 and 7, 3 and 7, 2 and 5 will never be together. The partition method won’t work.

In fact, the situation is similar to placing 4 similar balls in three similar boxes.

So we have to place 4 prime numbers in 3 places. If a place remains empty after distribution that means we will assume number 1 over there. Lets find the number of ways of distributing these prime numbers. Since the box are all same only the different grouping of number matters.

As we are talking about integers, the numbers of possible cases are:

· All positive

· One positive and two negative (we will have to find different ways of assigning positive and negative signs also in this case)