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Solve the inequality .^(n-1)C4 - .^(n -...

Solve the inequality
`.^(n-1)C_4 - .^(n -1)C_3 - 5/4 .^(n-2)C_2``lt 0,n in N`

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The number of positive integers satisfying the inequality .^(n+1)C_(n-2) - .^(n + 1)C_(n - 1) le 100 is

Prove that if ngt7 then .^(n-1)C_(3)+.^(n-1)C_(4)gt^(n)C_(3)

If .^(n)C_(4)=.^(n)C_(2) , then find .^(n)C_(3) .

If .^(n)C_(5)=.^(n)C_(3) , then find .^(n)C_(4) .

If .^(n)C_(5)=.^(n)C_(4) , then find .^(n)C_(7) .

Prove that .^(n)C_(0) +5 xx .^(n)C_(1) + 9 xx .^(n)C_(2) + "…." + (4n+1) xx .^(n)C_(n) = (2n+1) 2^(n) .

Prove that .^n C_0 .^n C_0-^(n+1)C_1 . ^n C_1+^(n+2)C_2 . ^n C_2- .. =(-1)^n .

Prove that ^nC_0 "^(2n)C_n-^nC_1 ^(2n-2)C_n +^nC_2 ^(2n-4)C_n =2^n

Prove that .^(n)C_(0) - .^(n)C_(1) + .^(n)C_(2) - .^(n)C_(3) + "……" + (-1)^(r) .^(n)C_(r) + "……" = (-1)^(r ) xx .^(n-1)C_(r ) .

Prove that .^(n)C_(1) - (1+1/2) .^(n)C_(2) + (1+1/2+1/3) .^(n)C_(3) + "…" + (-1)^(n-1) (1+1/2+1/3 + "…." + 1/n) .^(n)C_(n) = 1/n

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