Applying the Binomial Theorem - Algebra 2
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Find the value of $\binom{9}{4}$.
Find the value of $\binom{9}{4}$.
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$126$. Using $\binom{9}{4}=\frac{9!}{4!5!}=126$.
$126$. Using $\binom{9}{4}=\frac{9!}{4!5!}=126$.
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Expand $(x+y)^3$.
Expand $(x+y)^3$.
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$x^3+3x^2y+3xy^2+y^3$. Using coefficients $1,3,3,1$ from Pascal's triangle.
$x^3+3x^2y+3xy^2+y^3$. Using coefficients $1,3,3,1$ from Pascal's triangle.
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Expand $(x-y)^3$.
Expand $(x-y)^3$.
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$x^3-3x^2y+3xy^2-y^3$. Alternating signs from $(-y)^k$ with coefficients $1,3,3,1$.
$x^3-3x^2y+3xy^2-y^3$. Alternating signs from $(-y)^k$ with coefficients $1,3,3,1$.
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What is the coefficient of $x^5y^3$ in $(x-y)^8$?
What is the coefficient of $x^5y^3$ in $(x-y)^8$?
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$-\binom{8}{3}=-56$. Coefficient is $\binom{8}{3}=56$ with negative sign from $(-y)^3$.
$-\binom{8}{3}=-56$. Coefficient is $\binom{8}{3}=56$ with negative sign from $(-y)^3$.
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Find the value of $\binom{10}{1}$.
Find the value of $\binom{10}{1}$.
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$10$. Using $\binom{10}{1}=10$.
$10$. Using $\binom{10}{1}=10$.
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What is the factorial definition of the binomial coefficient $\binom{n}{k}$?
What is the factorial definition of the binomial coefficient $\binom{n}{k}$?
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$\binom{n}{k}=\frac{n!}{k!(n-k)!}$. Uses factorial formula to calculate combinations.
$\binom{n}{k}=\frac{n!}{k!(n-k)!}$. Uses factorial formula to calculate combinations.
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State the Binomial Theorem formula for expanding $(x+y)^n$ using binomial coefficients.
State the Binomial Theorem formula for expanding $(x+y)^n$ using binomial coefficients.
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$(x+y)^n=\sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^k$. The general formula for binomial expansion with coefficients and powers.
$(x+y)^n=\sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^k$. The general formula for binomial expansion with coefficients and powers.
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Expand $(x-y)^3$.
Expand $(x-y)^3$.
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$x^3-3x^2y+3xy^2-y^3$. Alternating signs from $(-y)^k$ with coefficients $1,3,3,1$.
$x^3-3x^2y+3xy^2-y^3$. Alternating signs from $(-y)^k$ with coefficients $1,3,3,1$.
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Expand $(x-y)^4$.
Expand $(x-y)^4$.
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$x^4-4x^3y+6x^2y^2-4xy^3+y^4$. Alternating signs from $(-y)^k$ with coefficients $1,4,6,4,1$.
$x^4-4x^3y+6x^2y^2-4xy^3+y^4$. Alternating signs from $(-y)^k$ with coefficients $1,4,6,4,1$.
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What is the coefficient of $x^7y^2$ in $(x+y)^9$?
What is the coefficient of $x^7y^2$ in $(x+y)^9$?
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$\binom{9}{2}=36$. The coefficient of $x^7y^2$ is $\binom{9}{2}=36$.
$\binom{9}{2}=36$. The coefficient of $x^7y^2$ is $\binom{9}{2}=36$.
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What is the coefficient of $x^3y^5$ in $(x+y)^8$?
What is the coefficient of $x^3y^5$ in $(x+y)^8$?
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$\binom{8}{5}=56$. The coefficient of $x^3y^5$ is $\binom{8}{5}=56$.
$\binom{8}{5}=56$. The coefficient of $x^3y^5$ is $\binom{8}{5}=56$.
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What is the coefficient of $x^6y^4$ in $(x+y)^{10}$?
What is the coefficient of $x^6y^4$ in $(x+y)^{10}$?
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$\binom{10}{4}=210$. The coefficient of $x^6y^4$ is $\binom{10}{4}=210$.
$\binom{10}{4}=210$. The coefficient of $x^6y^4$ is $\binom{10}{4}=210$.
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What is the coefficient of $x^5y^3$ in $(x-y)^8$?
What is the coefficient of $x^5y^3$ in $(x-y)^8$?
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$-\binom{8}{3}=-56$. Coefficient is $\binom{8}{3}=56$ with negative sign from $(-y)^3$.
$-\binom{8}{3}=-56$. Coefficient is $\binom{8}{3}=56$ with negative sign from $(-y)^3$.
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What is the coefficient of $x^4y^4$ in $(x-y)^8$?
What is the coefficient of $x^4y^4$ in $(x-y)^8$?
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$\binom{8}{4}=70$. Coefficient is $\binom{8}{4}=70$ with positive sign from $(-y)^4$.
$\binom{8}{4}=70$. Coefficient is $\binom{8}{4}=70$ with positive sign from $(-y)^4$.
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Identify the coefficient of the $x^2y^3$ term in $(x+y)^5$.
Identify the coefficient of the $x^2y^3$ term in $(x+y)^5$.
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$\binom{5}{3}=10$. The coefficient of $x^2y^3$ is $\binom{5}{3}=10$.
$\binom{5}{3}=10$. The coefficient of $x^2y^3$ is $\binom{5}{3}=10$.
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What is the coefficient of $a^4b^2$ in $(a+b)^6$?
What is the coefficient of $a^4b^2$ in $(a+b)^6$?
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$\binom{6}{2}=15$. The coefficient of $a^4b^2$ is $\binom{6}{2}=15$.
$\binom{6}{2}=15$. The coefficient of $a^4b^2$ is $\binom{6}{2}=15$.
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What is the coefficient of $a^2b^4$ in $(a+b)^6$?
What is the coefficient of $a^2b^4$ in $(a+b)^6$?
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$\binom{6}{4}=15$. The coefficient of $a^2b^4$ is $\binom{6}{4}=15$.
$\binom{6}{4}=15$. The coefficient of $a^2b^4$ is $\binom{6}{4}=15$.
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What is the coefficient pattern (Pascal row) for expanding $(x+y)^3$?
What is the coefficient pattern (Pascal row) for expanding $(x+y)^3$?
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$1,3,3,1$. Row 3 of Pascal's triangle.
$1,3,3,1$. Row 3 of Pascal's triangle.
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Find the value of $\binom{8}{2}$.
Find the value of $\binom{8}{2}$.
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$28$. Using $\binom{8}{2}=\frac{8!}{2!6!}=\frac{8 \cdot 7}{2}=28$.
$28$. Using $\binom{8}{2}=\frac{8!}{2!6!}=\frac{8 \cdot 7}{2}=28$.
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Find the value of $\binom{8}{6}$.
Find the value of $\binom{8}{6}$.
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$28$. Using symmetry: $\binom{8}{6}=\binom{8}{2}=28$.
$28$. Using symmetry: $\binom{8}{6}=\binom{8}{2}=28$.
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Find the value of $\binom{10}{1}$.
Find the value of $\binom{10}{1}$.
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$10$. Using $\binom{10}{1}=10$.
$10$. Using $\binom{10}{1}=10$.
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Find the value of $\binom{10}{2}$.
Find the value of $\binom{10}{2}$.
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$45$. Using $\binom{10}{2}=\frac{10 \cdot 9}{2}=45$.
$45$. Using $\binom{10}{2}=\frac{10 \cdot 9}{2}=45$.
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Find the value of $\binom{10}{3}$.
Find the value of $\binom{10}{3}$.
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$120$. Using $\binom{10}{3}=\frac{10 \cdot 9 \cdot 8}{6}=120$.
$120$. Using $\binom{10}{3}=\frac{10 \cdot 9 \cdot 8}{6}=120$.
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What is the coefficient sum of $(x-y)^n$ (equivalently, evaluate at $x=1,y=1$)?
What is the coefficient sum of $(x-y)^n$ (equivalently, evaluate at $x=1,y=1$)?
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$0$ if $n$ is odd; $2^n$ if $n$ is even. Substitute $x=1$ and $y=-1$ into $(x-y)^n$.
$0$ if $n$ is odd; $2^n$ if $n$ is even. Substitute $x=1$ and $y=-1$ into $(x-y)^n$.
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In $(x+y)^n$, what is the general term (the $k$th term) written in powers of $x$ and $y$?
In $(x+y)^n$, what is the general term (the $k$th term) written in powers of $x$ and $y$?
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$\binom{n}{k}x^{n-k}y^k$. The $(k+1)$th term in the binomial expansion.
$\binom{n}{k}x^{n-k}y^k$. The $(k+1)$th term in the binomial expansion.
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Find the value of $\binom{9}{4}$.
Find the value of $\binom{9}{4}$.
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$126$. Using $\binom{9}{4}=\frac{9!}{4!5!}=126$.
$126$. Using $\binom{9}{4}=\frac{9!}{4!5!}=126$.
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Expand $(x+y)^2$.
Expand $(x+y)^2$.
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$x^2+2xy+y^2$. Using coefficients $1,2,1$ from Pascal's triangle.
$x^2+2xy+y^2$. Using coefficients $1,2,1$ from Pascal's triangle.
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Expand $(x+y)^3$.
Expand $(x+y)^3$.
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$x^3+3x^2y+3xy^2+y^3$. Using coefficients $1,3,3,1$ from Pascal's triangle.
$x^3+3x^2y+3xy^2+y^3$. Using coefficients $1,3,3,1$ from Pascal's triangle.
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Expand $(x+y)^4$.
Expand $(x+y)^4$.
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$x^4+4x^3y+6x^2y^2+4xy^3+y^4$. Using coefficients $1,4,6,4,1$ from Pascal's triangle.
$x^4+4x^3y+6x^2y^2+4xy^3+y^4$. Using coefficients $1,4,6,4,1$ from Pascal's triangle.
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In $(x+y)^n$, what is the exponent of $x$ in the term containing $y^k$?
In $(x+y)^n$, what is the exponent of $x$ in the term containing $y^k$?
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$n-k$. Exponents of $x$ and $y$ must sum to $n$.
$n-k$. Exponents of $x$ and $y$ must sum to $n$.
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