Binomial Theorem

The number of integral solutions of x₁ + x₂ + x₃ = 0, with x_i $\ge$ - 5, is :

• ${}^{15}\mathrm{C}_2$

• ${}^{16}\mathrm{C}_2$

• ${}^{17}\mathrm{C}_2$

• ${}^{18}\mathrm{C}_2$

If ${}^{\mathrm{n}}\mathrm{C}_{\mathrm{r} - 1} = 36, {}^{\mathrm{n}}\mathrm{C}_{\mathrm{r}} = 84 \mathrm{and} {}^{\mathrm{n}}\mathrm{C}_{\mathrm{r} + 1} = 126$, then n is equal to

• 8

• 9

• 10

• 11

Coefficient of xⁿ in the expansion of $1 + \frac{\mathrm{a} + \mathrm{bx}}{1!} + \frac{{\left(\mathrm{a} + \mathrm{bx}\right)}^2}{2!} + \frac{{\left(\mathrm{a} + \mathrm{bx}\right)}^3}{3!} + ...$

• $\frac{{\mathrm{e}}^{\mathrm{a}} . {\mathrm{b}}^{\mathrm{n}}}{\mathrm{n}!}$

• $\frac{{\left(\mathrm{b} . \mathrm{a}\right)}^{\mathrm{n}}}{\mathrm{n}}$

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

• $\frac{{\mathrm{a}}^{\mathrm{n}} . {\mathrm{b}}^{\mathrm{n} - 1}}{\mathrm{n}!}$

The value of C₁ - 2 . C₂ + 3 . C₃ - 4 . C₄ + ... where C_r = ${}^{\mathrm{n}}\mathrm{C}_{\mathrm{r}}$ will be

• - 1

• 1

• 0

• None of these

The middle term in the expansion of ${\left(\frac{\mathrm{b}\sqrt{\mathrm{a}}}{5} - \frac{5}{\mathrm{a}\sqrt{\mathrm{b}}}\right)}^{12}$ is

• ${}^{12}\mathrm{C}_6{\left(\frac{\mathrm{b}}{\mathrm{a}}\right)}^3$

• $- {}^{12}\mathrm{C}_6{\left(\frac{\mathrm{b}}{\mathrm{a}}\right)}^3$

• ${}^{12}\mathrm{C}_7{\frac{\mathrm{b}}{\mathrm{a}}}^5$

• $- {}^{12}\mathrm{C}_7\frac{{\mathrm{b}}^5}{\mathrm{a}}$

The coefficient of x⁴ in the expansion of (1 + x + x² + x³)¹¹ is

• 990

• 605

• 810

• None of these

The coefficient of x⁴ in the expansion of log (1 + 3x + 2x²) is

• $\frac{16}{3}$

• $- \frac{16}{3}$

• $\frac{17}{4}$

• $- \frac{17}{4}$

If m = ${}^{\mathrm{n}}\mathrm{C}_2$, then ${}^{\mathrm{m}}\mathrm{C}_2$ is equal to

• $\mathrm{n} + {}^1\mathrm{C}_4$

• $3 \times {}^{\mathrm{n}}\mathrm{C}_4$

• $3 \times {}^{\mathrm{n} + 1}\mathrm{C}_4$

• None of these

The largest term in the expansion of (3 + 2x)⁵⁰ where x = $\frac{1}{5}$, is

• 7th

• 5th

• 8th

• 49th

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

• 210

• 100

• 310

• 110