ok guys these are some useful concepts which can help u solve a vast variety of problems on P & C ..

 

Lets start with a ques and then we can generalize the concept :

 

i.e. Let set A consist of n elements . Subsets P&Q are taken out of A with replacement. Find the no. of ways so that

i) PQ =  

=> In such q's we consider a paricular element of set A, say A₃

Now A₃ can be in either P or Q or in both or in neither (i.e. 4 cases).. looking at the above condition, the "both" case is not to be taken into consideration,

hence total no. of ways = 3*3* ...... n times

                                 => 3ⁿ ways

 

generalizing this q, in case of N no. of sets

Total no. of possibilities(for one element, say A₃) will be (2^N - 1) (for case (i) to be true)

=> because each element has two possibilities i.e. it can be either in a paricular set or not and -1 is the case when the element is in all sets ..

 

ii) Similarly ,

   for |PQ| = 1

Total no. of ways will be ⁿC₁ * 3^n-1  (guess u can figure out the logic urself .. )

 

Other Types (Some useful results on composite nos. )

 

Consider N = a^p * b^q * c^r  ..... where a,b,c .... are distinct primes

 

i) No. of divisors = (p+1) *(q+1)*(r+1) .......

 

and no. of proper divisors will be[ (i) - 2 ] (i.e. excluding 1 and the no. itself)

 

ii) Sum of divisors is

    (a^p+1 -1) /(a-1) * (b^q+1 -1)/(b-1) ..... (derived using concept of GP)

 

iii) No. of ways in which N can be resolved as a product of 2 factors is

       

    a) if N is not perfect square (i.e. if either p or q or r .... is odd)

=> no.of ways = 1/2 (p+1)(q+1)(r+1) ....

   

    b) If N is a perfect square (i.e. p and q and r .... are even )

=> no. of ways = 1/2 (p+1)(q+1)(r+1) ....... + or - 1

   + 1  when the leftover single factor is to be included

and - 1 if it is to be excluded ..

 

iv) No. of ways in which N can be resolved as a product of two co-prime factors is given by ^:

 => 2 ^n-1 ways .. where n is the no. of distinct primes

 

For expln of any of the 4 parts stated above , do nudge me ..

Anyways I'll continue with this essay later on ..

 

Hope u find this useful .. If u do plz do rate me ..

 

Thanks ..