Properties of binomial coefficients — practice questions

Question 1 · difficulty L2 · understanding

If A denotes the sum of all the coefficients in the expansion of $\left(1-3 x+10 x^2\right)^{\mathrm{n}}$ and B denotes the sum of all the coefficients in the expansion of $\left(1+x^2\right)^n$, then :

• A. $\mathrm{B}=\mathrm{A}^3$
• B. $3 \mathrm{A}=\mathrm{B}$
• C. $A=3 B$
• D. $\mathrm{A}=\mathrm{B}^3$

Question 2 · difficulty L2 · understanding

Let [ x ] denote greatest integer less than or equal to x. If for n$$\in$$N, $${(1 - x + {x^3})^n} = \sum\limits_{j = 0}^{3n} {{a_j}{x^j}}$$, then $$\sum\limits_{j = 0}^{\left[ {{{3n} \over 2}} \right]} {{a_{2j}} + 4} \sum\limits_{j = 0}^{\left[ {{{3n - 1} \over 2}} \right]} {{a_{2j}} + 1}$$ is equal to :

• A. 2 n $$-$$ 1
• B. n
• C. 2
• D. 1

Question 3 · difficulty L2 · understanding

The coefficient of $$x^{18}$$ in the expansion of $$\left(x^{4}-\frac{1}{x^{3}}\right)^{15}$$ is __________.

Question 4 · difficulty L2 · understanding

If the sum of the coefficients in the expansion of (x + y) n is 4096, then the greatest coefficient in the expansion is _____________.

Question 5 · difficulty L3 · understanding

If for $3 \leq r \leq 30$, $\left({^{30}C_{30-r}}\right) + 3\left({^{30}C_{31-r}}\right) + 3\left({^{30}C_{32-r}}\right) + \left({^{30}C_{33-r}}\right) = {^mC_r}$, then m equals :

• A. 31
• B. 32
• C. 33
• D. 34

Question 6 · difficulty L3 · understanding

The sum of the coefficients of $x^{499}$ and $x^{500}$ in $(1 + x)^{1000} + x(1 + x)^{999} + x^2(1 + x)^{998} + \ldots + x^{1000}$ is :

• A. ${ }^{1002} C_{501}$
• B. ${ }^{1001} C_{501}$
• C. ${ }^{1000} C_{501}$
• D. ${ }^{1002} C_{500}$

Question 7 · difficulty L3 · understanding

Let $\mathrm{S}=\frac{1}{25!}+\frac{1}{3!23!}+\frac{1}{5!21!}+\ldots$ up to 13 terms. If $13 \mathrm{~S}=\frac{2^k}{n!}, k \in \mathrm{~N}$, then $n+k$ is equal to

• A. 50
• B. 52
• C. 49
• D. 51

Question 8 · difficulty L3 · understanding

The sum of all possible values of $\mathbf{n} \in \mathbf{N}$, so that the coefficients of $x, x^2$ and $x^3$ in the expansion of $\left(1+x^2\right)^2(1+x)^{\mathrm{n}}$, are in arithmetic progression is :

• A. 12
• B. 9
• C. 3
• D. 7

Question 9 · difficulty L3 · understanding

The value of $\frac{{ }^{100} \mathrm{C}_{50}}{51}+\frac{{ }^{100} \mathrm{C}_{51}}{52}+\ldots .+\frac{{ }^{100} \mathrm{C}_{100}}{101}$ is:

• A. $\frac{2^{101}}{101}$
• B. $\frac{2^{100}}{101}$
• C. $\frac{2^{100}}{100}$
• D. $\frac{2^{101}}{100}$

Question 10 · difficulty L3 · understanding

The coefficient of $x^{48}$ in $(1+x)+2(1+x)^2+3(1+x)^3+\ldots+100(1+x)^{100}$ is equal to

• A. $100 \cdot{ }^{100} \mathrm{C}_{49}-{ }^{100} \mathrm{C}_{48}$
• B. $100 \cdot{ }^{101} \mathrm{C}_{49}-{ }^{101} \mathrm{C}_{50}$
• C. ${ }^{100} \mathrm{C}_{50}+{ }^{101} \mathrm{C}_{49}$
• D. $100 \cdot{ }^{100} \mathrm{C}_{49}-{ }^{100} \mathrm{C}_{50}$