You can find the coefficients of the required terms by using the Binomial Theorem. Alternatively, use the fact that the Pascal's triangle will be symmetric.
False, see solution.
Practice makes perfect
By using the Binomial Theorem we can write the given expression using sigma notation.
(x+y)^(20) = ∑_(k=0)^(20) 20!/k!(20-k)!x^(20-k)y^kThe eighth term of the expression is found when k=7 and the twelfth term is the one when k=11. Let's substitute k=7 to find the eighth coefficient.
As we can see, the eighth and the twelfth terms have different coefficients. Thus, the given statement is false.
Alternative Solution
Alternative Solution
We can know the coefficients of (x+y)^(20) by finding the 20th row of the Pascal's triangle. Since the coefficients of the given binomial are both equal to 1, the coefficients in the triangle will be symmetric.
In the20th row, there are21terms.
The middle term will be the one in the 11th position and the row will be symmetric with respect to this coefficient.
From the above, we conclude that the eighth and the twelfth terms will have different coefficients. Thus, the given statement is false.
