List of top Mathematics Questions on Binomial Theorem for Positive Integral Indices

Using Binomial Theorem, evaluate \((102)^5\).
Evaluate\( ( √3 +√2) ^6 - ( √3 -√2) ^6.\)
Expand the expression \((1- 2x)^ 5\).
Expand the expression\( (\frac{2}{x} - \frac{x}{2})^5\).
Expand the expression \((2x - 3) ^6\).
Expand the expression \((\frac{x}{3} + \frac{1}{x})^5\).
Expand the expression \((x+ \frac{1}{x})^6\).
Using Binomial Theorem, evaluate \((96)^3\).
Using Binomial Theorem, evaluate \((101)^4\)
Using Binomial Theorem, evaluate \((99)^5\).
Using Binomial Theorem, indicate which number is larger \((1.1)^{10000}\) or \( 1000.\)
Find \((a + b)^4 - (a - b)^4\). Hence, evaluate \((\sqrt3 + \sqrt2)^4 - (\sqrt3 - \sqrt2)^4\).

If a and b are distinct integers, prove that a - b is a factor of \(a^n - b^n\) , whenever n is a positive integer. 
[Hint: write\( a ^n = (a - b + b)^n\) and expand]

Find \((x + 1)^6 - (x - 1)^6\). Hence or otherwise evaluate \((\sqrt2 + 1)^6 + (\sqrt2 - 1)^6.\)
Show that \(9^{ n+1} -8n -9\) is divisible by \(64\), whenever n is a positive integer.
Prove that \(\underset{r=0} {\overset{n}∑} { 3^r }\,{ ^nC_r} = 4^n\)