The coefficient of x⁴ in (1 + x + x³ + x⁴)¹⁰ is

• 210

• 100

• 310

• 110

$\mathrm{The} \mathrm{coefficient} \mathrm{of}{ }^{ }{\mathrm{x}}^4 \mathrm{in} \mathrm{the} \mathrm{expansion} \mathrm{of} \frac{{\left(1 - 3\mathrm{x}\right)}^2}{\left(1 - 2\mathrm{x}\right)} \mathrm{is} \mathrm{equal} \mathrm{to}$

• 1

• 2

• 3

• 4

$$\begin{gathered} \mathrm{If} {\left(1 + \mathrm{x}\right)}^{\mathrm{n}} = {\mathrm{C}}_0 + {\mathrm{C}}_1\mathrm{x} + {\mathrm{C}}_2{\mathrm{x}}^2 + .... + {\mathrm{C}}_{\mathrm{n}}{\mathrm{x}}^{\mathrm{n}} ,\mathrm{then} \\ {\mathrm{C}}_0 + 2{\mathrm{C}}_1 + 3{\mathrm{C}}_2 + ..... + \left(\mathrm{n} + 1\right){\mathrm{C}}_{\mathrm{n} }\mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• $2^{\mathrm{n}} + \mathrm{n} 2^{\mathrm{n} - 1}$

• $2^{\mathrm{n}} + \mathrm{n} 2^{\mathrm{n} }$

• $2^{\mathrm{n}} + \left(\mathrm{n} + 1\right)2^{\mathrm{n} }$

• $2^{\mathrm{n} - 1} + \left(\mathrm{n} - 1\right)2^{\mathrm{n} }$

$$\begin{gathered} \left|\mathrm{x}\right| < 1,\mathrm{the} \mathrm{coefficient} \mathrm{of} {\mathrm{x}}^3 \mathrm{in} \mathrm{the} \mathrm{expansion} \mathrm{of} \\ \log \left(1 + \mathrm{x} + {\mathrm{x}}^2\right) \mathrm{in} \mathrm{ascendingnpowers} \mathrm{of} \mathrm{x},\mathrm{is} \end{gathered}$$

• $\frac{2}{3}$

• $\frac{4}{3}$

• $- \frac{2}{3}$

• $- \frac{4}{3}$

65.

The least value of the natural number n satisfying C(n, 5) + C(n, 6) > C(n + 1, 5)

• 10

• 11

• 12

• 13

The sum of the coefficients in the expansion of (1 + x + x²)ⁿ is

• 2

• 2ⁿ

• 3ⁿ

• 4

In the expansion of (1 + x)ⁿ the coefficients of pth and (p + 1) th terms are respectively p and q, then p + q is equal to

• n

• n + 1

• n + 2

• n + 3

If $\frac{1 - \mathrm{x} + 6{\mathrm{x}}^2}{1 - {\mathrm{x}}^3} = \frac{\mathrm{A}}{\mathrm{x}} + \frac{\mathrm{B}}{1 + \mathrm{x}} + \frac{\mathrm{C}}{1 + \mathrm{x}}$, then A is equal to

• 1

• 2

• 3

• 4

If the coefficient of (2r + 1)th term and (r + 2)th term in the expansion of (1 + x)⁴³ are equal, then r is equal to :

• 12

• 14

• 16

• 18

The coefficient of x⁵ in the expansion of (1 + x²)⁵(1 + x)⁴, is

• 60

• 50

• 40

• 56