Hi. We are doing permutations and combinations in class and we were given some formulas without proof to remember. I was able to derive most of them but was unable to derive 3 of them. But I would like to see how do I derive them for sake of fun (also if I forget them what will I do. :) ).(adsbygoogle = window.adsbygoogle || []).push({});

1. A number is expressed in the form of product of it's prime factors

N = a^x b^y c^z etc.

Then the number N can be resolved as a product of 2 factors in how many ways?

[Ans : If N is not a perfect square 0.5(x+1)(y+1)(z+1)

If N is a perfect square 0.5[(x+1)(y+1)(z+1)+1]

]

All I can think is to take it to be of this form xy = N.

Then what do I do next.

2. This one is similar to first. The number of ways in which a composite number N can be resolved into 2 factors which are co prime to each other is 2^(n-1) , where n is number of different factors in N (eg: as in last case n=x+y+z)

3. Number of ways of arranging n differnet objects in r boxes where arrangement matters within a box

[(n+r-1)!] / [(r-1)!]

Any help?

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# Some combinatorics proofs

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