`30.^(30)C_(0)+29.^(30)C_(1)+ . . . +1.^(30)C_(29)=m2^n and m , n in N ` then m+n =

If m, n, r, in N then .^(m)C_(0).^(n)C_(r) + .^(m)C_(1).^(n)C_(r-1)+"…….."+.^(m)C_(r).^(n)C_(0) = coefficient of x^(r) in (1+x)^(m)(1+x)^(n) = coefficient of x^(f) in (1+x)^(m+n) The value of r(0 le r le 30) for which S = .^(20)C_(r).^(10)C_(0) + .^(20)C_(r-1).^(10)C_(1) + ........ + .^(20)C_(0).^(10)C_(r) is minimum can not be

If (1+a)^(n)=.^(n)C_(0)+.^(n)C_(1)a+.^(n)C_(2)a^(2)+ . . +.^(n)C_(n)a^(n) , then prove that .^(n)C_(1)+2.^(n)C_(2)+3.^(n)3C_(3)+ . . .+n.^(n)C_(n)=n.2^(n-1) .

If the value of the sum 29(.^(30)C_(0))+28(.^(30)C_(1))+27(.^(30)C_(2))+…….+1(.^(30)C_(28))+0.(.^(30)C_(29))-(.^(30)C_(30)) is equal to K.2^(32) , then the value of K is equal to

If ^(n)C_(12)=^(n)C_(8) then = a.20 b.12 c.6 d.30

If m, n, r, in N then .^(m)C_(0).^(n)C_(r) + .^(m)C_(1).^(n)C_(r-1)+"…….."+.^(m)C_(r).^(n)C_(0) = coefficient of x^(r) in (1+x)^(m)(1+x)^(n) = coefficient of x^(f) in (1+x)^(m+n) The value of r for which S = .^(20)C_(r.).^(10)C_(0)+.^(20)C_(r-1).^(10)C_(1)+"........".^(20)C_(0).^(10)C_(r) is maximum can not be

Let m ,n in N and g c d ( 2, n)=1. if 30((30),(0))+((30),(1))+....+2((30),(28))+1((30),(29))=n.2^m then n+m is equal to ( here ((n),(k)) =""^(n)C_K )

If .^(m)C_(1)=.^(n)C_(2) prove that m=(1)/(2)n(n-1) .

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that C_(0)^(2) - C_(1)^(2) + C_(2)^(2) -…+ (-1)^(n) *C_(n)^(2)= 0 or (-1)^(n//2) * (n!)/((n//2)! (n//2)!) , according as n is odd or even Also , evaluate C_(0)^(2) + C_(1)^(2) + C_(2)^(2) - ...+ (-1)^(n) *C_(n)^(2) for n = 10 and n= 11 .