Question 1198624: Find the 7th term of the binomial (x+y)^9.
One of the following is the correct answer. Which one?
A) 84x^3 y^6
B) 36x^2 y^7
C) 210x^3 y^6
D) 84x^2 y^7
E) 36x^6 y^3
Found 3 solutions by math_tutor2020, Theo, MathTherapy:
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
There are n+1 = 9+1 = 10 terms total.
The first term has x^9y^0.
The tenth term has x^0y^9.
The term number k leads to the y having an exponent of k-1.
Therefore, the 7th term will have y^(7-1) = y^6 involved. This immediately leads to x^3
So we have x^3y^6 as the entire variable term.
Note the exponents add to n = 9.
The coefficient is found by looking at the 7th item in the row 1,9,36,... in Pascal's Triangle. That value is 84.
Alternatively, use the nCr combination formula with n = 9 and r = 6.
We use r = 6 instead of r = 7 because the count starts at r = 0. Meaning r = 6 is the 7th term.
Therefore, we end up with 84x^3y^6 as the 7th term of (x+y)^9.
Use software like WolframAlpha to confirm this.
Answer: Choice A
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! i believe it's selection B which is 36x^2y^7
the general formula is (x + y) ^ n = sum (c(n,k) * x^(n-k) * y^k) from k = 0 to n.
each term is c(n,k) * x^(n-k) * y^k
when n = 9, .....
the first term is c(9,0) * x^9 * y^0
the second term is c(9,1) * x^8 * y^1
the third term is c(9,2) * x^7 * y^2
the fourth term is c(9,3) * x^6 * y^3
the fifth term is c(9,4) * x^5 * y^4
the sixth term is c(9,5) * x^4 * y^5
the seventh term is c(9,6) * x^3 * y^6
the eighth term is c(9,7) * x^2 * y^7
the 9th term is c(9,8) * x^1 * y^8
the 10th term is c(9,9) * x^0 * y^9
you are looking for the seventh term.
that term is c(9,6) * x^3 * y^6
c(9,6) = 9! / (6! * 3!) = (9*8*7*6!) = (6!*3!) = (9*8*7)/(3*2*1) = 84
the seventh term is therefore 84 * x^3 * y^6.
that would be selection A.
i confirmed with an online calculator.
the results of that calculator are shown below:
the terms are what's between the + signs.
the seventh term is therefore equal to 84x^3y^6
Answer by MathTherapy(10552)  (Show Source):
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