The binomial coefficients which are in decreasing order are

• ${}^{15}{\mathrm{C}}_{5,}{ }^{15}{\mathrm{C}}_{6,}{ }^{15}{\mathrm{C}}_{7,}$

• ${}^{15}{\mathrm{C}}_{10} ,{ }^{15}{\mathrm{C}}_9 ,{ }^{15}{\mathrm{C}}_8$

• ${}^{15}{\mathrm{C}}_6 ,{ }^{15}{\mathrm{C}}_7 ,{ }^{15}{\mathrm{C}}_8$

• ${}^{15}{\mathrm{C}}_7 ,{ }^{15}{\mathrm{C}}_6 ,{ }^{15}{\mathrm{C}}_5$

$$\begin{gathered} \mathrm{If} \frac{\mathrm{x} - 4}{{\mathrm{x}}^2 -5\mathrm{x} + 6} \mathrm{can} \mathrm{be} \mathrm{expanded} \mathrm{in} \mathrm{the} \mathrm{ascending} \mathrm{power} \mathrm{of} \mathrm{x}, \\ \mathrm{then} \mathrm{the} \mathrm{coefficient} \mathrm{of} {\mathrm{x}}^3 \mathrm{is} \end{gathered}$$

• $- \frac{73}{648}$

• $\frac{73}{648}$

• $\frac{71}{648}$

• $- \frac{71}{648}$

Coefficient of x¹⁰ in the expansion of (2 + 3x)e^{- x} is

• $\frac{- 26}{\left(10\right)!}$

• $\frac{- 28}{\left(10\right)!}$

• $\frac{- 30}{\left(10\right)!}$

• $\frac{- 32}{\left(10\right)!}$

If (1 + x)¹⁵ = a₀ + a₁x + ... + a₁₅x¹⁵, then $\sum _{\mathrm{r} = 1}^{15}\mathrm{r}\frac{{\mathrm{a}}_{\mathrm{r}}}{{\mathrm{a}}_{\mathrm{r}} - 1}$ is equal to

• 110

• 115

• 120

• 135

The coefficient of x³y⁴z⁵ in the expansion of (xy + yz + xz)⁶ is

• 70

• 60

• 50

• None of these

If $\left|\mathrm{x}\right| < \frac{1}{2}$, then the coefficient of x^r in the expansion of $\frac{1 +\hspace{0.17em}2\mathrm{x}}{{\left(1 - 2\mathrm{x}\right)}^2}$, is

• r2^r

• (2r - 1)2^r

• r2^{2r + 1}

• (2r + 1)2^r

77.

$\mathrm{The} \mathrm{coefficient} \mathrm{of} {\mathrm{x}}^{\mathrm{n}} \mathrm{in} \frac{1 - 2\mathrm{x}}{{\mathrm{e}}^{\mathrm{x}}} \mathrm{is} :$

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

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

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

• ${\left(- 1\right)}^{\mathrm{n}}\frac{\left(1 + 4\mathrm{n}\right)}{\mathrm{n}!}$

$\mathrm{If} {\mathrm{a}}_{\mathrm{k}} \mathrm{is} \mathrm{the} \mathrm{coefficient} \mathrm{of} {\mathrm{x}}^{\mathrm{k}} \mathrm{in} \mathrm{the} \mathrm{expansion} \mathrm{of} {\left(1 + \mathrm{x} + {\mathrm{x}}^2\right)}^{\mathrm{n}} \mathrm{for} \mathrm{k} = 0, 1, 2, . . , 2\mathrm{n}, \mathrm{then}$

• - a₀

• 3ⁿ

• n 3^{n + 1}

• n 3ⁿ

The coefficient of x^k in the expansion of $\frac{1 - 2\mathrm{x} - {\mathrm{x}}^2}{{\mathrm{e}}^{- \mathrm{x}}}$ is

• $\frac{1 - \mathrm{k} - {\mathrm{k}}^2}{\mathrm{k}!}$

• $\frac{{\mathrm{k}}^2 + 1}{\mathrm{k}!}$

• $\frac{1 - \mathrm{k}}{\mathrm{k}!}$

• $\frac{1}{\mathrm{k}!}$

The coefficient of x²⁴ in the expansion of (1 + x²)¹²(1 + x¹²)(1 + x²⁴) is

• ${}_{12}\mathrm{C}_6$

• ${}_{12}\mathrm{C}_6 + 2$

• ${}_{12}\mathrm{C}_6 + 4$

• ${}_{12}\mathrm{C}_6 +\hspace{0.17em}6$