Podcast
Questions and Answers
What is the sixth term in the expansion of $(w - 2)^{10}$?
What is the sixth term in the expansion of $(w - 2)^{10}$?
Identify the middle term in the expansion of $(3x^2 - 2x)^6$.
Identify the middle term in the expansion of $(3x^2 - 2x)^6$.
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 $( oot{y} + y^2)^{10}$?
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 $k$?
Signup and view all the answers
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 $( oot{x} - x^2)^{10}$?
Signup and view all the answers
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 $x$?
Signup and view all the answers
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?
Signup and view all the answers
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 $(2z + 3x)^6$?
Signup and view all the answers
What is the value of $n$ if $nC10 = nC5$?
What is the value of $n$ if $nC10 = nC5$?
Signup and view all the answers
If $2nC3/nC2 = 12$, what is the value of $n$?
If $2nC3/nC2 = 12$, what is the value of $n$?
Signup and view all the answers
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 $m$?
Signup and view all the answers
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 $r$?
Signup and view all the answers
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 $(2 - 2x)^{10}$?
Signup and view all the answers
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 $a$?
Signup and view all the answers
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 $k$?
Signup and view all the answers
What is the fifth term in the expansion of $(2z - y)^8$?
What is the fifth term in the expansion of $(2z - y)^8$?
Signup and view all the answers
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?
Signup and view all the answers
What is the general term in the expansion of (a + b)n?
What is the general term in the expansion of (a + b)n?
Signup and view all the answers
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?
Signup and view all the answers
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?
Signup and view all the answers
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?
Signup and view all the answers
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?
Signup and view all the answers
What does 8C3 represent in the calculation of the coefficient of x5?
What does 8C3 represent in the calculation of the coefficient of x5?
Signup and view all the answers
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?
Signup and view all the answers
What is the value of $nC0$ for any integer $n$?
What is the value of $nC0$ for any integer $n$?
Signup and view all the answers
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 $(a + b)^n$?
Signup and view all the answers
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?
Signup and view all the answers
Which identity holds true according to the properties of binomial coefficients?
Which identity holds true according to the properties of binomial coefficients?
Signup and view all the answers
How is the factorial of zero defined?
How is the factorial of zero defined?
Signup and view all the answers
What is the result of applying the Binomial Theorem to $(a+b)^n$?
What is the result of applying the Binomial Theorem to $(a+b)^n$?
Signup and view all the answers
Which of the following correctly expresses $(a + b)^n$ in expanded form?
Which of the following correctly expresses $(a + b)^n$ in expanded form?
Signup and view all the answers
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?
Signup and view all the answers
What is the relationship denoted by $nC_r + nC_{r-1} = nC_{r+1}$?
What is the relationship denoted by $nC_r + nC_{r-1} = nC_{r+1}$?
Signup and view all the answers
What is the coefficient of $x^9y^3$ in the expression derived?
What is the coefficient of $x^9y^3$ in the expression derived?
Signup and view all the answers
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?
Signup and view all the answers
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?
Signup and view all the answers
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 $(a + b)^n$?
Signup and view all the answers
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?
Signup and view all the answers
How many terms are there in the binomial expansion of $(a + b)^n$?
How many terms are there in the binomial expansion of $(a + b)^n$?
Signup and view all the answers
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 $(3 + 9y)^{10}$?
Signup and view all the answers
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?
Signup and view all the answers
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.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Related Documents
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.
