If in the expansion of (a + b)ⁿ and (a + b)^{n + 3}, the ratio of the coefficients of second and third terms and third and fourth terms respectively are equal then n is _____.

If in the expansion of (1 +x)ⁿ, the coefficients of fifth, sixth and seventh terms are inA.P. then n is equal to _____.

The middle term in the expansion of \(\left( \frac { { 2x }^{ 2 } }{ 3 } +\frac { 3 }{ { 2x }^{ 2 } } \right) ^{ 10 }\)is _____.

The coefficient of x⁵y⁸ in the expansion of (x + y)¹³ is _____.

Constant term in the expansion of \(\left( x-\frac { 1 }{ x } \right) ^{ 14 }\) _____.

Find (a + b)⁴ - (a - b)⁴. Hence, evaluate \({ \left( \sqrt { 3 } +\sqrt { 2 } \right) }^{ 4 }-{ \left( \sqrt { 3 } -\sqrt { 2 } \right) }^{ 4 }\)

Simplify the following expression
\({ \left( y+\frac { 1 }{ y } \right) }^{ 6 }-{ \left( y-\frac { 1 }{ y } \right) }^{ 6 }\)

Find the middle term(s) in the given expansion.
\(\left( \frac { x }{ 3 } +9y \right) ^{ 10 }\)

If P and Q are the sum of odd and even terms in the expansion\(\left( y+b \right) ^{ n }\), then prove that\(\left( y+b \right) ^{ 2n }+\left( y-b \right) ^{ 2n }=2\left( { P }^{ 2 }+Q^{ 2 } \right) \).

Find the coefficient of x in the expansion of (1-3x+7x²) (1-x)¹⁶

If the coefficients of x⁷ and x⁸in the expansion of \(\left( 3+\frac { x }{ { 2 } } \right) ^{ n }\)

The coefficients of three consecutive terms in the expansion of (1+x)ⁿ .

Find (a + b)⁵ - (a - b)⁵. Hence evaluate \(\left( \sqrt { 5 } +\sqrt { 3 } \right) ^{ 5 }+\left( \sqrt { 5 } -\sqrt { 3 } \right) ^{ 5 }\)

In the binomial expansion of (1 + x)ⁿ, the coefficient of the 5^th, 6^th and 7^th terms are in A.P. Find all values of n for which this can happen.

The 3^rd, 4^th and 5^th terms in the expansion of (x +a)ⁿ are 84, 280 and 560 respectively. Find the values of x, a and n.

Evaluate the following terms. General term in the expansion of (3x² + 4y)¹⁰ .

Expand \(\left( \frac { 2x }{ 3 } -\frac { 3 }{ 2x } \right) ^{ 4 }\)
Use \(\left( { a-b } \right) ^{ n }=^{ n }{ C }_{ o }{ a }^{ n }-^{ n }{ C }_{ 1 }{ a }^{ n-1 }b+^{ n }{ C }_{ 2 }{ a }^{ n-2 }{ b }^{ 2 }+..+(-1)^{ n }\quad ^{ n }{ C }_{ n }{ b }^{ n }\)

Expand \(\left( 102 \right) ^{ 5 }\)

Find the coefficient of x6 in the expansion \(\left( x-\frac { { 1 } }{ 6^2 } \right) ^{ 24 }\) .

Find the term independent of x in the expansion of \(\left( 2x+\frac { 3 }{ { x }^{ 2 } } \right) ^{ 9 }\)

Find the middle terms in the expansion of \(\left( 2x-\frac { { x }^{ 2 } }{ 6 } \right) ^{ 9 }\)

In the expansion of (x + a)ⁿ, sums of odd and even terms are P and Q respectively, Prove that
(i) 2 (P² + Q²) = (x + a)²ⁿ + (x - a)²ⁿ
(ii) p² - Q² = (x² - a²)ⁿ

The coefficient of (m + 1) ^th term in the expansion of (1 +x)²ⁿ is equal to the coefficient of (m + 3)^th term. Show that m + 1 = n.