Prove that ²ⁿC_{n =}  \(\frac { { 2 }^{ n }\times 1\times3\times ...(2n-1) }{ n! } \)

Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly three aces in each combination.

Prove 1.3 + 2.3² + 3.3³+...+n-3ⁿ=\(\frac{(2n-1)3^{n+1}+3}{4}\) for all n ∈ N

Prove that the sum of the first n non-zero even numbers is n² + n,

n² - n is divisible by 6, for each natural number n \(\ge\) 2.

1 + 5 + 9 + ... + (4n - 3) = n(2n -1), \(\forall\)n \(\in\)N.

Find the sum of the series \(\frac { { 1 }^{ 3 } }{ 1 } +\frac { { 1 }^{ 3 }+{ 2 }^{ 3 } }{ 1+3 } +\frac { { 1 }^{ 3 }+{ 2 }^{ 3 }+3^{ 3 }+.... }{ 1+3+5 } to\ n\ terms\)

Find the coefficient of x in the expansion of \(log(\frac{1}{1-5x+6x^2})\).

Expand \({1\over(1+3x)^2} \) in powers of x. Find a condition on x for which the expansion is valid.

Find the value of \((a^{2}+\sqrt{a^{2}-1})^{4}+(a^{2}-\sqrt{a^{2}-1})^{4}\)