Question
What number does C3 represent in the expansion
? Explain.
The correct answer is: = 4
ANSWER:
Hint:
The binomial expansion is ,
.
The value of ![n C subscript r equals fraction numerator n factorial over denominator r factorial left parenthesis n minus r right parenthesis factorial end fraction](data:image/png;base64,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)
We are asked to find what number does
represent in the expansion .
![C subscript 0 a to the power of 5 plus C subscript 1 a to the power of 4 b plus C subscript 2 a squared b squared plus C subscript 3 a b cubed plus C subscript 4 b to the power of 4.](data:image/png;base64,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)
Step 1 of 2:
The expansion is: ![C subscript 0 a to the power of 5 plus C subscript 1 a to the power of 4 b plus C subscript 2 a squared b squared plus C subscript 3 a b cubed plus C subscript 4 b to the power of 4](data:image/png;base64,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)
Here,
represents the coefficient of in the expansion.
Step 2 of 2:
The value of n=4, so the value of
is:
![4 C subscript 3 equals fraction numerator 4 factorial over denominator 3 factorial left parenthesis 4 minus 3 right parenthesis factorial end fraction](data:image/png;base64,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)
![equals fraction numerator 4 factorial over denominator 3 factorial cross times 1 factorial end fraction](data:image/png;base64,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)
= 4
Note:
The value
is called the combination permutation.
Related Questions to study
Dakota said the third term of the expansions of
.Explain Dakota’s error and then correct the answer.
You can use Both the Pascal’s triangle and binomial expansion to find the value of (x + y)n .
Dakota said the third term of the expansions of
.Explain Dakota’s error and then correct the answer.
You can use Both the Pascal’s triangle and binomial expansion to find the value of (x + y)n .
What number does C3 represent in the expansion
? Explain.
The value is called the combination permutation.
What number does C3 represent in the expansion
? Explain.
The value is called the combination permutation.
Explain how to use a polynomial identity to factor .![8 x to the power of 6 minus 27 y cubed](data:image/png;base64,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)
Explain how to use a polynomial identity to factor .![8 x to the power of 6 minus 27 y cubed](data:image/png;base64,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)
Use polynomial identities to factor the polynomials or simplify the expressions :
![10 cubed minus 3 cubed](data:image/png;base64,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)
Use polynomial identities to factor the polynomials or simplify the expressions :
![10 cubed minus 3 cubed](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAD8AAAARCAYAAABq+XSZAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAQ3ZOC+gAAAXBJREFUeNpjYKAN+A/Fv4D4AhC7MQwcGFC3aADxQ4bBAejuFiYgfjdIPE9XtwgD8RQgThwEHifoFjUgroXmDVyAD4inAfFJKJ4JFcOV12Jp5BlHIN4GxN+A+AcQ3wLiCVBPkuWWxUCcBlWIC+wFYj8kvg9UDFdAtULNpDbYD8RBQMyGJGYAxDsodQsuz4cC8SQs4qAQj8Rj3g86Ju9PBOR/kOv5NTiqCkeoHDZgC8Rn6ORxJSC+gUeeKLfg8vw7tGQGAyCxl1jy2A9oMpSnsafFoQUZKN974cjvRLsFl+f/4NHzawAbMDBcQC1DSfX8Dyp5ABsmBASBOABa+9jTyvMvcSR7DrRkP1CABxoANPE8qFDzwCLuhqfAozf4QSvPg+rV2VjE5xOo6ugF9KCFHk08D2tcgAoWFmgWqAbigwPg0YPQdgcLtM0Oqm7vA3E8AxULH2yFy1xo8voGbd7yDIDnQXX2Fqg7fkAjxYdhFBAHADADba9Tb0OXAAAAk3RFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtc3VwPjxtbj4xMDwvbW4+PG1uPjM8L21uPjwvbXN1cD48bW8+JiN4MjIxMjs8L21vPjxtc3VwPjxtbj4zPC9tbj48bW4+MzwvbW4+PC9tc3VwPjwvbWF0aD7v3q5xAAAAAElFTkSuQmCC)
How are Pascal’s triangle and binomial expansion such as
related?
How are Pascal’s triangle and binomial expansion such as
related?
Explain how to use a polynomial identity to factor
.
We use identities to speed up the process of multiplication and simplification. There are some basic polynomial identities that you need to by heart.
Explain how to use a polynomial identity to factor
.
We use identities to speed up the process of multiplication and simplification. There are some basic polynomial identities that you need to by heart.
Use polynomial identities to factor the polynomials or simplify the expressions :
![10 cubed plus 5 cubed](data:image/png;base64,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)
Use polynomial identities to factor the polynomials or simplify the expressions :
![10 cubed plus 5 cubed](data:image/png;base64,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)
Explain why the middle term
is 10x.
Explain why the middle term
is 10x.
Use polynomial identities to factor the polynomials or simplify the expressions :
![9 cubed plus 6 cubed](data:image/png;base64,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)
Use polynomial identities to factor the polynomials or simplify the expressions :
![9 cubed plus 6 cubed](data:image/png;base64,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)
How can you use polynomial identities to rewrite expressions efficiently ?
How can you use polynomial identities to rewrite expressions efficiently ?
Use binomial theorem to expand![left parenthesis 2 c plus d right parenthesis to the power of 6](data:image/png;base64,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)
Use binomial theorem to expand![left parenthesis 2 c plus d right parenthesis to the power of 6](data:image/png;base64,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)
Use polynomial identities to factor the polynomials or simplify the expressions :
![open parentheses 1 over 16 x to the power of 6 minus 25 y to the power of 4 close parentheses](data:image/png;base64,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)
Use polynomial identities to factor the polynomials or simplify the expressions :
![open parentheses 1 over 16 x to the power of 6 minus 25 y to the power of 4 close parentheses](data:image/png;base64,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)
Use binomial theorem to expand .![left parenthesis x minus 1 right parenthesis to the power of 7](data:image/png;base64,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)
Use binomial theorem to expand .![left parenthesis x minus 1 right parenthesis to the power of 7](data:image/png;base64,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)
Use polynomial identities to factor the polynomials or simplify the expressions :
![64 x cubed minus 125 y to the power of 6](data:image/png;base64,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)
Use polynomial identities to factor the polynomials or simplify the expressions :
![64 x cubed minus 125 y to the power of 6](data:image/png;base64,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)
How are Pascal’s triangle and binomial expansion such as (a + b)5 related?
You can find the expansion of (x + y)n using both Pascal’s triangle and binomial expansion.
How are Pascal’s triangle and binomial expansion such as (a + b)5 related?
You can find the expansion of (x + y)n using both Pascal’s triangle and binomial expansion.