Question 1

Which entry in Pascal's Triangle corresponds to the binomial coefficient $$\displaystyle \binom{7}{4}$$?

Accepted Answer

The entry in row 7, position 4 (counting from 0).

Question 2

What is the value of $$\displaystyle \binom{6}{5} as $$shown in Pascal's Triangle?

Accepted Answer

$$6$$

Question 3

How many shortest lattice paths are there from $$(0,0) to$$ $$(3,2) on $$the integer lattice?

Accepted Answer

$$\displaystyle \binom{5}{2} = 10$$

Question 4

Which formula correctly expresses the symmetry property of binomial coefficients?

Accepted Answer

$$\displaystyle \binom{n}{k} = \binom{n}{n-k}$$

Question 5

If you want to find the number of ways to choose 3 objects from 15, which binomial coefficient do you use?

Accepted Answer

$$\displaystyle \binom{15}{3}$$

Question 6

What is the value of $$\displaystyle \binom{15}{3}$$?

Accepted Answer

$$455$$

Question 7

Which of the following best describes a lattice path from $$(0,0) to$$ $$(3,2)$$?

Accepted Answer

A sequence of 3 right moves and 2 up moves, in any order.

Question 8

Which property of Pascal's Triangle allows you to compute $$\displaystyle \binom{n}{k}$$ using the two entries above it?

Accepted Answer

$$\displaystyle \binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$$

Question 9

Suppose you want to walk from $$(0,0) to$$ $$(4,3)$$ using only the shortest lattice paths. How many such paths are there?

Accepted Answer

$$\displaystyle \binom{7}{3} = 35$$

Question 10

Which of the following is NOT a property of Pascal's Triangle?

Accepted Answer

Each entry is the product of the two entries above it.

Question 11

A real-world application: In a hospital, a nurse must visit 5 rooms on the first floor and 3 rooms on the second floor. If the nurse must visit all rooms, but can choose the order, how many different shortest routes are there if the nurse must go from the first to the second floor only once?

Accepted Answer

$$\displaystyle \binom{8}{3} = 56$$

Question 12

If you expand $$(x+y)^6$$, what is the coefficient of $$x^2y^4$?

Accepted Answer

$$\displaystyle \binom{6}{4} = 15$$

Question 13

Which of the following expressions gives the number of ways to arrange 4 right moves and 2 up moves in a lattice path?

Accepted Answer

$$\displaystyle \binom{6}{2}$$

Question 14

If you want to find the number of subsets of a set with 7 elements, which sum would you use from Pascal's Triangle?

Accepted Answer

$$2^7 = 128$

Question 15

Which of the following is the correct formula for the binomial coefficient?

Accepted Answer

$$\displaystyle \binom{n}{k} = \frac{n!}{k!(n-k)!}$$

Pascal's Triangle and Lattice Paths: Foundations for Discrete Models and Combinatorics

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