If m parallel lines in a plane are intersected by a family of n parallel lines. Find the number of parallelogram formed?

If the fourth term in the expansion of \({ \left( ax+\frac { 1 }{ x } \right) }^{ n }\) is \(\frac { 5 }{ 2 } \)  then find the values of a and n.

Using the principle of mathematical induction, prove that 1.3 + 2.3² + 3.3³ + ... + n.3ⁿ =\(\frac { (2n-1){ 3 }^{ n+1 }+3 }{ 4 } for\ all\ n\in N\)

Using binomial theorem, find the value of \({ \left( \sqrt { 2 } +1 \right) }^{ 5 }+{ \left( \sqrt { 2 } -1 \right) }^{ 5 }\)

Find the 11^th term from the end in \({ \left( 2x-\frac { 1 }{ { x }^{ 2 } } \right) }^{ 25 }\)