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Find the sum of the coefficients of two middle terms in the binomial expansion of  \[\left( 1 + x \right)^{2n - 1}\]

Write the total number of terms in the expansion of  \[\left( x + a \right)^{100} + \left( x - a \right)^{100}\] .

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

Find the centroid of a triangle, mid-points of whose sides are (1, 2, –3), (3, 0, 1) and (–1, 1, –4). 

If A and B are the sums of odd and even terms respectively in the expansion of (x + a)ⁿ, then (x + a)²ⁿ − (x − a)²ⁿ is equal to

The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B are (3, –5, 7) and (–1, 7, –6) respectively, find the coordinates of the point C.

The number of irrational terms in the expansion of \[\left( 4^{1/5} + 7^{1/10} \right)^{45}\]  is

In the expansion of \[\left( x^2 - \frac{1}{3x} \right)^9\] , the term without x is equal to

If the distance between the points P(a, 2, 1) and Q (1, −1, 1) is 5 units, find the value of a. 

If an the expansion of \[\left( 1 + x \right)^{15}\]   , the coefficients of \[\left( 2r + 3 \right)^{th}\text{  and  } \left( r - 1 \right)^{th}\]  terms are equal, then the value of r is

Write the coordinates of third vertex of a triangle having centroid at the origin and two vertices at (3, −5, 7) and (3, 0, 1). 

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

If in the expansion of \[\left( x^4 - \frac{1}{x^3} \right)^{15}\] ,  \[x^{- 17}\]  occurs in rth term, then

In the expansion of \[\left( x - \frac{1}{3 x^2} \right)^9\]  , the term independent of x is

If in the expansion of (1 + y)ⁿ, the coefficients of 5th, 6th and 7th terms are in A.P., then nis equal to

In the expansion of \[\left( \frac{1}{2} x^{1/3} + x^{- 1/5} \right)^8\] , the term independent of x is

If the sum of odd numbered terms and the sum of even numbered terms in the expansion of  \[\left( x + a \right)^n\]  are A and B respectively, then the value of \[\left( x^2 - a^2 \right)^n\] is 

The total number of terms in the expansion of \[\left( x + a \right)^{100} + \left( x - a \right)^{100}\]  after simplification is

The middle term in the expansion of \[\left( \frac{2x}{3} - \frac{3}{2 x^2} \right)^{2n}\] is 

If r^th term is the middle term in the expansion of \[\left( x^2 - \frac{1}{2x} \right)^{20}\]  then \[\left( r + 3 \right)^{th}\]  term is