Prove the following: `P(n , n)=P(n , n-1)`

Prove the following: P(n , n)=2P(n , n-2)

Prove the following: P(n , r)=P(n-1, r)+rdotP(n-1,\ r-1)

Prove the following: P(n , r)=ndotP(n-1,\ r-1)

Find the value of n from each of the following: (i) 10 .^(n)P_(6) = .^(n+1)P_(3) (ii) 16.^(n)P_(3) = 13 .^(n+1)P_(3)

Prove that P(n,n) = P(n,n-1)

Prove that : P(n,n)= 2P (n,n -2)

Evaluate the following (i) lim_(n to oo)((1)/(n^(2))+(2)/(n^(2))+(3)/(n^(2))....+(n-1)/(n^(2))) (ii) lim_(n to oo)((1)/(n+1)+(1)/(n+2)+....+(1)/(2n)) (iii) lim_(n to oo)((n)/(n^(2)+1^(2))+(n)/(n^(2)+2^(2))+....+(n)/(2n^(2))) (iv) lim_(n to oo)((1^(p)+2^(p)+.....+n^(p)))/(n^(p+1)),pgt0

Prove that P(n,r) = (n- r+1) P(n,r-1)

If P(n ,5): P(n ,3)=2:1 find n

Define a function phi:NtoN as follows phi(1)=1,phi(P^(n))=P^(n-1)(P-1) is prime and n epsilonN and phi(mn)=phi(m)phi(n) if m & n are relatively prime natural numbers. The number of natural numbers n such that phi(n) is odd is