Show that: ^mC_1+^(m+1)C_2+^(m+2)C_3+…….+^(m+n-1)C_n=^nC_1+^(n+1)C_2+^(n+2)C_3+…+^(n+m-1)C_m`.

Prove that ^m C_1^n C_m-^m C_2^(2n)C_m+^m C_3^(3n)C_m-=(-1)^(m-1)n^mdot

Using binomial theorem (without using the formula for .^n C_r ) , prove that .^nC_4+^m C_2-^m C_1.^n C_2 = .^m C_4-^(m+n)C_1.^m C_3+^(m+n)C_2.^m C_2-^(m+n)C_3^m.C_1 +^(m+n)C_4dot

If m in N and mgeq2 prove that: |1 1 1\ ^m C_1\ ^(m+1)C_1\ ^(m+2)C_1\ ^m C_2\ ^(m+1)C_2\ ^(m+2)C_2|=1 .

The value of .^(n)C_(1)+.^(n+1)C_(2)+.^(n+2)C_(3)+"….."+.^(n+m-1)C_(m) is equal to a. .^(m+n)C_(n) - 1 b. .^(m+n)C_(n-1) c. .^(m)C_(1) + ^(m+1)C_(2) + ^(m+2)C_(3) + "…." + ^(m+n-1)C_(n) d. .^(m+n)C_(m) - 1

Prove that "^n C_0^(2n)C_n-^n C_1^(2n-1)C_n+^n C_2xx^(2n-2)C_n++(-1)^n^n C_n^n C_n=1.

The value of ^n C_1+^(n+1)C_2+^(n+2)C_3++^(n+m-1)C_m is equal to

Show that sum_(k=m)^n ^kC_r=^(n+1)C_(r+1)-^mC_(r+1)

Prove that .^(n)C_(1) + 2 xx .^(n)C_(2) + 3 xx .^(n)C_(3) + "…." + n xx .^(n)C_(n) = n2^(n-1) . Hence, prove that .^(n)C_(1).(.^(n)C_(2))^(2).(.^(n)C_(3))^(3)"......."(.^(n)C_(n))^(n) le ((2^(n))/(n+1))^(.^(n+1)C_(2)) AA n in N .

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n)," prove that " 1^(2)*C_(1) + 2^(2) *C_(2) + 3^(2) *C_(3) + …+ n^(2) *C_(n) = n(n+1)* 2^(n-2) .

The value of |1 1 1^n C_1^(n+2)C_1^(n+4)C_1^n C_2^(n+2)C_2^(n+4)C_2| is