Question 1

Evaluate the binomial coefficient

Accepted Answer

A

Question 2

Evaluate the binomial coefficient

Accepted Answer

B

Question 3

Evaluate the binomial coefficient

Accepted Answer

C

Question 4

Evaluate the binomial coefficient

Accepted Answer

The answer of Evaluate the binomial coefficient

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Question 5

Evaluate the binomial coefficient

Accepted Answer

The answer of Evaluate the binomial coefficient

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Question 6

Expand (x - 2)⁵using the binomial theorem.

Accepted Answer

The binomial theorem allows for the expansion of $(x - 2)^{15}$. The signs alternate starting with a positive sign because the negative term is raised to an even power for the first term, and the coefficients follow the pattern of Pascal's triangle or can be calculated using combinations. The correct expansion thus starts with $x^{15}$ and alternates signs with decreasing powers of $x$, following the binomial coefficients for $n=15$.

Question 7

Expand (4x + 5)³ using the binomial theorem.

Accepted Answer

Using the binomial theorem, (4x + 5)^3 = (4x)^3 + 3(4x)^2(5) + 3(4x)(5)^2 + (5)^3 = 64x^3 + 240x^2 + 300x + 125.

Question 8

Expand  using the binomial theorem.

Accepted Answer

The answer of Expand  using the binomial theorem.
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Question 9

Expand (3x - y)⁴ using Pascal's Triangle.

Accepted Answer

The correct expansion using Pascal's Triangle coefficients (1, 14, 91, 364, 1001, 2002, 3003, 3432, 3003, 2002, 1001, 364, 91, 14, 1) and the binomial theorem results in alternating signs due to the negative term (-y), starting with a positive sign for the highest degree of x term.

Question 10

Expand (5x + 3y)³ using Pascal's Triangle.

Accepted Answer

To expand $(5x + 3y)^3$ using Pascal's Triangle, we use the coefficients from the row corresponding to $n=3$, which are 1, 3, 3, 1. The expansion is $(5x)^3 + 3(5x)^2(3y) + 3(5x)(3y)^2 + (3y)^3$, simplifying to $125x^3 + 225x^2y + 135xy^2 + 27y^3$.

Question 11

Find the coefficient, C, of the term in the binomial expansion. Binomial: (x + 5)⁹ Term: Cx⁶

Accepted Answer

The term in the binomial expansion is given by the formula: C(n, k) * a^(n-k) * b^k. Here, n=9, a=x, b=5, and we need the term with x^6, so k=3. The coefficient C is C(9, 3) * 5^3 = 84 * 125 = 10,500.

Question 12

Find the coefficient, C, of the term in the binomial expansion. Binomial: (x - 4y)⁶Term: Cx³y³

Accepted Answer

The binomial expansion of (x - 4y)^6 is given by the formula: C(n, k) * (x)^(n-k) * (-4y)^k. For the term with x^3y^3, n = 6 and k = 3. The coefficient is C(6, 3) * (-4)^3 = 20 * (-64) = -1,280.

Question 13

Find the coefficient, C, of the term in the binomial expansion. Coefficient: (5x + 2y)⁸Term: Cx⁵y³

Accepted Answer

The answer of Find the coefficient, C, of the term...

Question 14

Find the coefficient, C, of the term in the binomial expansion. Binomial: (4x - 3y)⁴ Term: Cxy³

Accepted Answer

The coefficient, C, of the term $xy^{13}$ in the expansion of $(4x - 3y)^{14}$ can be found using the binomial theorem, which in general form is $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k$. For the term $xy^{13}$, $n=14$, $k=13$, $a=4x$, and $b=-3y$. Thus, the coefficient is $\binom{14}{13} (4x)^{14-13}(-3y)^{13} = 14 \cdot 4 \cdot (-3)^{13}$. Calculating this gives $-432$, as the negative sign comes from the odd power of -3.

Question 15

Evaluate the binomial coefficient.

Accepted Answer

The answer of Evaluate the binomial coefficient.
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Question 16

Expand (2x + 5)⁴ using the binomial theorem.

Accepted Answer

The answer of Expand (2x + 5)⁴ using the binomial...

Question 17

In a state lottery in which 5 numbers are drawn from a possible 53 numbers, the number of possible 5 number combinations is equal to:
  How many possible combinations are there?

Accepted Answer

The answer of In a state lottery in which 5...

Question 18

Expand the expression using the binomial theorem.

Accepted Answer

The answer of Expand the expression using the binomial theorem....

Question 19

Expand the expression using the binomial theorem.

Accepted Answer

The answer of Expand the expression using the binomial theorem....

Question 20

Evaluate the binomial coefficient

Accepted Answer

The answer of Evaluate the binomial coefficient 
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Quiz 10: Sequences and Series

Evaluate the binomial coefficient

Evaluate the binomial coefficient

Evaluate the binomial coefficient

Evaluate the binomial coefficient

Evaluate the binomial coefficient

Expand (x - 2) ⁵using the binomial theorem.

Expand (4x + 5) ³ using the binomial theorem.

Expand using the binomial theorem.

Expand (3x - y) ⁴ using Pascal's Triangle.

Expand (5x + 3y) ³ using Pascal's Triangle.

Find the coefficient, C, of the term in the binomial expansion. Binomial: (x + 5) ⁹ Term: Cx⁶

Find the coefficient, C, of the term in the binomial expansion. Binomial: (x - 4y) ⁶Term: Cx³y³

Find the coefficient, C, of the term in the binomial expansion. Coefficient: (5x + 2y) ⁸Term: Cx⁵y³

Find the coefficient, C, of the term in the binomial expansion. Binomial: (4x - 3y) ⁴ Term: Cxy³

Evaluate the binomial coefficient.

Expand (2x + 5)⁴ using the binomial theorem.

In a state lottery in which 5 numbers are drawn from a possible 53 numbers, the number of possible 5 number combinations is equal to: How many possible combinations are there?

Expand the expression using the binomial theorem.

Expand the expression using the binomial theorem.

Evaluate the binomial coefficient

Quiz 10: Sequences and Series

Evaluate the binomial coefficient

Evaluate the binomial coefficient

Evaluate the binomial coefficient

Evaluate the binomial coefficient

Evaluate the binomial coefficient

Expand (x - 2) 5 using the binomial theorem.

Expand (4x + 5) 3 using the binomial theorem.

Expand using the binomial theorem.

Expand (3x - y) 4 using Pascal's Triangle.

Expand (5x + 3y) 3 using Pascal's Triangle.

Find the coefficient, C, of the term in the binomial expansion. Binomial: (x + 5) 9 Term: Cx6

Find the coefficient, C, of the term in the binomial expansion. Binomial: (x - 4y) 6 Term: Cx3y3

Find the coefficient, C, of the term in the binomial expansion. Coefficient: (5x + 2y) 8 Term: Cx5y3

Find the coefficient, C, of the term in the binomial expansion. Binomial: (4x - 3y) 4 Term: Cxy3

Evaluate the binomial coefficient.

Expand (2x + 5)4 using the binomial theorem.

In a state lottery in which 5 numbers are drawn from a possible 53 numbers, the number of possible 5 number combinations is equal to: How many possible combinations are there?

Expand the expression using the binomial theorem.

Expand the expression using the binomial theorem.

Evaluate the binomial coefficient

Expand (x - 2) ⁵using the binomial theorem.

Expand (4x + 5) ³ using the binomial theorem.

Expand (3x - y) ⁴ using Pascal's Triangle.

Expand (5x + 3y) ³ using Pascal's Triangle.

Find the coefficient, C, of the term in the binomial expansion. Binomial: (x + 5) ⁹ Term: Cx⁶

Find the coefficient, C, of the term in the binomial expansion. Binomial: (x - 4y) ⁶Term: Cx³y³

Find the coefficient, C, of the term in the binomial expansion. Coefficient: (5x + 2y) ⁸Term: Cx⁵y³

Find the coefficient, C, of the term in the binomial expansion. Binomial: (4x - 3y) ⁴ Term: Cxy³

Expand (2x + 5)⁴ using the binomial theorem.