Question 1

Which of the following correctly represents the Binomial Theorem for expanding $$(a + b)^n$$?

Accepted Answer

$$ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a$$^{n-k}$$ b^k $$

Question 2

What is the coefficient of $$x^3y^2$ in the expansion of (x$$ + $$y)^5$?

Accepted Answer

$$ 10 $$

Question 3

A fair coin is flipped 8 times. What is the probability that it lands on heads exactly 3 times? (Express your answer as a fraction.)

Accepted Answer

$$ \dfrac{56}{256} $$

Question 4

Which term in the expansion of $$(2x - 3)^6$ contains x^4$?

Accepted Answer

$$ 15 \cdot (2x)^4$ \cdot (-3)^2$ = 15 \cdot 16x^4$ \cdot 9 = 2160x^4$ $$

Question 5

In the expansion of $$(a + b$$)^{10}$$, what is the sum of all the coefficients?

Accepted Answer

$$ 1024 $$

Question 6

A company produces circuit boards with a 50% success rate. If 7 boards are produced, what is the probability that exactly 5 are correct? (Use the Binomial Theorem.)

Accepted Answer

$$ \dfrac{21}{128} $$

Question 7

Which of the following statements about Pascal’s Triangle is TRUE?

Accepted Answer

Each row is symmetric and the numbers read the same left to right as right to left.

Question 8

Find the fourth term in the expansion of $$(3x + 2)^5$.

Accepted Answer

$$ 80x^2$ $$

Question 9

A test consists of 10 true-false questions. If a student guesses on every question, what is the probability that they get exactly 8 correct answers?

Accepted Answer

$$ \dfrac{45}{1024} $$

Question 10

Which of the following is the correct expansion of $$(x - y)^3$ using Pascal’s Triangle?

Accepted Answer

$$ x^3$ - 3x^2$y + 3xy^2$ - y^3$ $$

Question 11

In the expansion of $$(a + b$$)^{12}$$, which term contains a^5$?

Accepted Answer

$$ \binom{12}{7} a^5$ b^7$ $$

Question 12

If $$(x + y)^n is $$expanded, how many terms will there be in the expansion?

Accepted Answer

$$ n + 1 $$

Question 13

A group of 8 contestants is selected from an audience with equal numbers of men and women. What is the probability that exactly 5 women and 3 men are selected?

Accepted Answer

$$ \dfrac{56}{256} $$

Question 14

Which shortcut can be used to write out the rows of Pascal’s Triangle without adding the numbers above?

Accepted Answer

Use the formula $$ \binom{n}{k} = \dfrac{n!}{k!(n-k)!} $$ for each entry in the row.

Question 15

Expand $$(2c - 6d)^4$ and select the correct coefficient for the cd^3$ term.

Accepted Answer

$$ -1728 $$

Question 16

If the probability of an event is not 50%, how can the Binomial Theorem be adapted to find the probability of $$k$$ successes in $$n$$ trials?

Accepted Answer

Substitute the actual probabilities for $$a$$ and $$b in$$ $$(a + b)^n$$ and use the coefficient of the desired term.

Question 17

What is the sixth term in the expansion of $$(\sqrt{a} + \sqrt{b}$$)^{12}$$?

Accepted Answer

$$ \binom{12}{5} (\sqrt{a}$$)^{7}$$ (\sqrt{b}$$)^{5}$$ $$

Question 18

Which of the following best explains why the coefficients in the expansion of $$(a + b)^n$$ are symmetric?

Accepted Answer

Because $$\binom{n}{k} = \binom{n}{n-k}$$, so the coefficients are the same for terms equidistant from the ends.

Question 19

In the expansion of $$(x + y)^n$$, what is the exponent of $$y in $$the $$kth $$term (starting from $$k=0$$)?

Accepted Answer

$$ k $$

Powers of Binomials: Pascal’s Triangle and the Binomial Theorem

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