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Questions and Answers
What is the sixth term in the expansion of $(w - 2)^{10}$?
What is the sixth term in the expansion of $(w - 2)^{10}$?
- 2520w^5
- 720w^2
- 1200w^3
- 2100w^4 (correct)
Identify the middle term in the expansion of $(3x^2 - 2x)^6$.
Identify the middle term in the expansion of $(3x^2 - 2x)^6$.
- 432x^5
- 540x^6
- 486x^4 (correct)
- 720x^2
How can you find the term free from $y$ in the expansion of $(
oot{y} + y^2)^{10}$?
How can you find the term free from $y$ in the expansion of $( oot{y} + y^2)^{10}$?
- Multiply by the constant term only
- Use the binomial theorem
- Calculate each term until the $y$ terms cancel out
- Set the exponent of $y$ to zero (correct)
If the coefficients of $a^7$ and $a^8$ in the expansion of $(2 + 3)^k$ are the same, what can be concluded about $k$?
If the coefficients of $a^7$ and $a^8$ in the expansion of $(2 + 3)^k$ are the same, what can be concluded about $k$?
Which term is independent of $x$ in the expansion of $(
oot{x} - x^2)^{10}$?
Which term is independent of $x$ in the expansion of $( oot{x} - x^2)^{10}$?
In the expansion of $(3x - 2x^3)^8$, which term is independent of $x$?
In the expansion of $(3x - 2x^3)^8$, which term is independent of $x$?
What is the middle term in the expansion of $(2 + 2x)^8$ if it equals 1120?
What is the middle term in the expansion of $(2 + 2x)^8$ if it equals 1120?
How do you find the coefficient of $x^5$ in the expansion of $(2z + 3x)^6$?
How do you find the coefficient of $x^5$ in the expansion of $(2z + 3x)^6$?
What is the value of $n$ if $nC10 = nC5$?
What is the value of $n$ if $nC10 = nC5$?
If $2nC3/nC2 = 12$, what is the value of $n$?
If $2nC3/nC2 = 12$, what is the value of $n$?
In the equation $15Cm = 15C(m + 1)$, what can be concluded about $m$?
In the equation $15Cm = 15C(m + 1)$, what can be concluded about $m$?
If $nCr/n-1C(r-1)$, what can be derived for $n$ in terms of $r$?
If $nCr/n-1C(r-1)$, what can be derived for $n$ in terms of $r$?
What is the coefficient of $x^{22}$ in the expansion of $(2 - 2x)^{10}$?
What is the coefficient of $x^{22}$ in the expansion of $(2 - 2x)^{10}$?
In the expression $21Ca = 21C(a + 1)$, what does this imply about $a$?
In the expression $21Ca = 21C(a + 1)$, what does this imply about $a$?
If the term independent of $x$ in the expansion of $(x^3 + x^8)$ equals 1320, what can be inferred about $k$?
If the term independent of $x$ in the expansion of $(x^3 + x^8)$ equals 1320, what can be inferred about $k$?
What is the fifth term in the expansion of $(2z - y)^8$?
What is the fifth term in the expansion of $(2z - y)^8$?
What values are assigned to a, b, and n in the expression (x + 3)8?
What values are assigned to a, b, and n in the expression (x + 3)8?
What is the general term in the expansion of (a + b)n?
What is the general term in the expansion of (a + b)n?
How do you determine the coefficient of x5 in the expansion of (x + 3)8?
How do you determine the coefficient of x5 in the expansion of (x + 3)8?
In the expansion of (x - x)14, what is the condition for a term to be independent of x?
In the expansion of (x - x)14, what is the condition for a term to be independent of x?
What is the value of r when finding the term independent of x in (x - x)14?
What is the value of r when finding the term independent of x in (x - x)14?
What is the result of -14C7 in the context of the expansion of (x - x)14?
What is the result of -14C7 in the context of the expansion of (x - x)14?
What does 8C3 represent in the calculation of the coefficient of x5?
What does 8C3 represent in the calculation of the coefficient of x5?
What is the final calculated coefficient of x5 in the expansion of (x + 3)8?
What is the final calculated coefficient of x5 in the expansion of (x + 3)8?
What is the value of $nC0$ for any integer $n$?
What is the value of $nC0$ for any integer $n$?
Which of the following represents the General Term in the Binomial Expansion of $(a + b)^n$?
Which of the following represents the General Term in the Binomial Expansion of $(a + b)^n$?
For the expression $(x - 2y)^{12}$, what is the coefficient of the fourth term?
For the expression $(x - 2y)^{12}$, what is the coefficient of the fourth term?
Which identity holds true according to the properties of binomial coefficients?
Which identity holds true according to the properties of binomial coefficients?
How is the factorial of zero defined?
How is the factorial of zero defined?
What is the result of applying the Binomial Theorem to $(a+b)^n$?
What is the result of applying the Binomial Theorem to $(a+b)^n$?
Which of the following correctly expresses $(a + b)^n$ in expanded form?
Which of the following correctly expresses $(a + b)^n$ in expanded form?
In the expression for the Binomial Theorem, the summation notation $\sum_{r=0}^{n} nC_r a^{n-r} b^r$ ranges over which variable?
In the expression for the Binomial Theorem, the summation notation $\sum_{r=0}^{n} nC_r a^{n-r} b^r$ ranges over which variable?
What is the relationship denoted by $nC_r + nC_{r-1} = nC_{r+1}$?
What is the relationship denoted by $nC_r + nC_{r-1} = nC_{r+1}$?
What is the coefficient of $x^9y^3$ in the expression derived?
What is the coefficient of $x^9y^3$ in the expression derived?
In the binomial expansion of $(a + b)^n$, how is the middle term identified if n is odd?
In the binomial expansion of $(a + b)^n$, how is the middle term identified if n is odd?
For the expansion of $(3 + 9y)^{10}$, what is the value of n?
For the expansion of $(3 + 9y)^{10}$, what is the value of n?
What is the formula for the $(r + 1)^{th}$ term in the expansion of $(a + b)^n$?
What is the formula for the $(r + 1)^{th}$ term in the expansion of $(a + b)^n$?
What is the value of $T_6$ when calculated using the values provided?
What is the value of $T_6$ when calculated using the values provided?
How many terms are there in the binomial expansion of $(a + b)^n$?
How many terms are there in the binomial expansion of $(a + b)^n$?
What is the relationship between the values of a and b in the expression $(3 + 9y)^{10}$?
What is the relationship between the values of a and b in the expression $(3 + 9y)^{10}$?
For an even value of n, what is the formula for determining the middle term?
For an even value of n, what is the formula for determining the middle term?
Flashcards
Factorial Notation
Factorial Notation
A way to express a product of consecutive integers from 1 to a given positive integer n. It's symbolized as n!”.
nCr
nCr
Represents the number of ways to choose r items from a set of n items without regard to order.
Binomial Theorem
Binomial Theorem
A formula that expands an expression raised to a positive integer power.
General Term in Binomial Expansion
General Term in Binomial Expansion
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Binomial Expression
Binomial Expression
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0! or 1!
0! or 1!
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nCx = nCy
nCx = nCy
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Binomial Expansion formula
Binomial Expansion formula
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Fourth term in (x - 2y)^12
Fourth term in (x - 2y)^12
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nCr = nCn-r
nCr = nCn-r
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Middle Term in Binomial Expansion (even n)
Middle Term in Binomial Expansion (even n)
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Middle Term in Binomial Expansion (odd n)
Middle Term in Binomial Expansion (odd n)
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rth Term in Binomial Expansion
rth Term in Binomial Expansion
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Binomial Coefficient (nCr)
Binomial Coefficient (nCr)
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Coefficient
Coefficient
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Term
Term
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Expansion of Binomials
Expansion of Binomials
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Binomial Theorem
Binomial Theorem
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Finding the coefficient of x^5 in (x+3)^8
Finding the coefficient of x^5 in (x+3)^8
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Term independent of x in (x-1/x)^14
Term independent of x in (x-1/x)^14
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Binomial Theorem
Binomial Theorem
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General term in binomial expansion
General term in binomial expansion
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nCr (Combination)
nCr (Combination)
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Coefficient
Coefficient
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Term independent of a variable
Term independent of a variable
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Finding the coefficient of x^n
Finding the coefficient of x^n
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nCx = nCy
nCx = nCy
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Combination Formula
Combination Formula
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Binomial Coefficient
Binomial Coefficient
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Term Independent of x
Term Independent of x
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Finding the 5th term
Finding the 5th term
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Coefficient of x^22
Coefficient of x^22
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Coefficient of y^-8
Coefficient of y^-8
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General Term
General Term
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Sixth term in (𝑤 − 2)^10
Sixth term in (𝑤 − 2)^10
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Middle term in (3𝑥^2 − 2𝑥)^6
Middle term in (3𝑥^2 − 2𝑥)^6
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Term free from y in (√𝑦 + 𝑦^2)^10 = 405k
Term free from y in (√𝑦 + 𝑦^2)^10 = 405k
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Equal coefficients of a^7 and a^8 in (2 + 3)^k
Equal coefficients of a^7 and a^8 in (2 + 3)^k
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Independent term of 𝑥 in (√𝑥 − 𝑥^2)^10
Independent term of 𝑥 in (√𝑥 − 𝑥^2)^10
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Independent term of 𝑥 in (3𝑥 + 1/√𝑥)^10
Independent term of 𝑥 in (3𝑥 + 1/√𝑥)^10
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Independent term of 𝑥 in (2𝑥 − 1/3𝑥)^10
Independent term of 𝑥 in (2𝑥 − 1/3𝑥)^10
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Independent term in (3𝑥 − 2𝑥^3)^8.𝛼
Independent term in (3𝑥 − 2𝑥^3)^8.𝛼
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Study Notes
Factorial Notation
- n! = n × (n − 1) × (n − 2) × ... × 3 × 2 × 1
- 0! = 1
- 1! = 1
Binomial Theorem
- General form: (a + b)n
- Expansion of (a + b)n for n ∈ N: nCo anb0 + nC1 an-1b1 + nC2 an-2b2 + ... + nCn-1 a1bn-1 + nCn a0bn or, ∑nr=0 nCr an-rbr
- nCr = n! / (r! (n − r)!)
- nC0 = nCn = 1, nC1 = n
- nCr = nCn-r
- nCr + nCr-1 = n + 1Cr
Binomial Theorem for Positive Integers
- Theorem: If a and b are real numbers, and n ∈ N: (a + b)n = nCo anb0 + nc1 an-1b1 + nc2 an-2b2 + ... + ncn-1 a1bn-1 + ncn a0bn
General Term in a Binomial Expansion of (a + b)n
- (r + 1)th term: Tr+1 = nCr an-rbr
Middle Terms in a Binomial Expansion of (a + b)n
- If n is even, the middle term is (n/2 + 1)th term
- If n is odd, the middle terms are (n+1)/2 and (n+3)/2th terms.
Additional Problems and Examples
- Various examples are provided for finding specific terms, coefficients, and the term independent of x in binomial expansions
- Examples and problems involve finding the value of n or other parameters in binomial expansions, given specific conditions.
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Description
Test your understanding of the Binomial Theorem and Factorial Notation with this quiz. It covers essential formulas, properties of binomial coefficients, and expansions pertinent to positive integers. Perfect for students diving into combinatorics and algebra concepts.
