Multiple Choice

1. What are the numbers in the 4th row of Pascal's Triangle, assuming the top row (Row 0) is a single '1'?

   1, 4, 6, 4, 1

   1, 5, 10, 10, 5, 1

   1, 3, 3, 1

   1, 2, 1

2. What is the sum of all numbers in the 6th row of Pascal's Triangle (where the first row, '1', is Row 0)?

   7

   32

   128

   64

3. Which of the following statements about Pascal's Triangle is true?

   The numbers in any row are always increasing from left to right.

   The sum of the numbers in any row is $n+1$, where $n$ is the row number.

   Each row is symmetrical; the numbers read the same from left to right as from right to left.

   Each number is the product of the two numbers directly above it.

4. The first row of Pascal's Triangle is Row 0. What is the value of the 3rd element (starting count from 0) in the 5th row?

   6

   10

   15

   5

5. The entries in Pascal's Triangle can be represented by combinatorial notation $\binom{n}{k}$. Which notation correctly represents the 4th element (index $k=3$) in the 6th row (index $n=6$)?

   $\binom{5}{3}$

   $\binom{7}{3}$

   $\binom{6}{4}$

   $\binom{6}{3}$

6. Calculate the value of $\binom{7}{3}$.

   21

   7

   35

   42

7. If Row 0 is the single '1' at the top of Pascal's Triangle, what is the value of the second term (index 1) in the 8th row (index 8)?

   7

   28

   1

   8

8. Using Pascal's Triangle, what are the coefficients for the expansion of $(a+b)^3$?

   1, 2, 1

   1, 4, 6, 4, 1

   1, 5, 10, 10, 5, 1

   1, 3, 3, 1

9. What is the coefficient of the term $a^2b^2$ in the expansion of $(a-b)^4$?

   -4

   4

   6

   -6

10. Expand $(2x+y)^3$ using the Binomial Theorem or Pascal's Triangle.

    $2x^3 + 6x^2y + 6xy^2 + y^3$

    $8x^3 + 4x^2y + 2xy^2 + y^3$

    $8x^3 + 12x^2y + 6xy^2 + y^3$

    $8x^3 + 12x^2y + 4xy^2 + y^3$

11. What is the general formula for the $(r+1)$-th term in the binomial expansion of $(a+b)^n$?

    $\binom{n}{r} a^r b^{n-r}$

    $\binom{n}{r+1} a^{n-(r+1)} b^{r+1}$

    $\binom{n}{k} a^{n-k} b^k$ (where $k$ is the term number, not the index)

    $\binom{n}{r} a^{n-r} b^r$

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

    $20x^2y^3$

    $10x^3y^2$

    $80x^2y^3$

    $40x^3y^2$

13. What is the coefficient of $x^2$ in the expansion of $(2x-3)^4$?

    216

    216

    12

    -72

14. Find the constant term in the expansion of $(x^2 + \frac{1}{x})^6$.

    1

    20

    6

    15

15. What is the sum of the coefficients in the expansion of $(x+y)^7$?

    128

    7

    14

    64

16. What is the sum of the coefficients in the expansion of $(3x-2y)^4$?

    0

    16

    1

    81

17. In Pascal's Triangle, if you start from the '1' at the top (Row 0) and follow the diagonal row of numbers immediately adjacent to the leading '1's, what type of numbers do you encounter? (e.g., 1, 2, 3, 4, ...)

    Counting numbers (natural numbers)

    Triangular numbers

    Fibonacci numbers

    Powers of 2

18. If $\binom{n}{2} = 28$, what is the value of $n$?

    8

    9

    7

    6

19. What is the expansion of $(x-1)^4$?

    $x^4 + 4x^3 - 6x^2 + 4x - 1$

    $x^4 + 4x^3 + 6x^2 + 4x + 1$

    $x^4 - 4x^3 + 6x^2 - 4x + 1$

    $x^4 - 4x^3 + 6x^2 - 4x - 1$

20. Which of the following statements is true regarding binomial coefficients?

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

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

    $\binom{n}{k} = \frac{n!}{k!}$

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

21. In Pascal's Triangle, if the second element (not counting the initial '1') of a certain row is 10, what is the row number (assuming the top '1' is Row 0)?

    Row 10

    Row 11

    Row 12

    Row 9

22. What is the coefficient of $x^4y^2$ in the expansion of $(x+2y)^6$?

    60

    15

    90

    30

23. How many terms are there in the expansion of $(a+b)^n$?

    $n-1$

    $2n$

    $n+1$

    $n$

24. What is the sum of the numbers in the 5th row of Pascal's Triangle, where the top '1' is considered Row 0?

    64

    16

    32

    5

25. Using the binomial expansion for $(1+x)^3$, what is the value of $(1.01)^3$ to two decimal places?

    1.02

    1.03

    1.04

    1.00