91Ó°ÊÓ

Pascal's Triangle is a triangular array of numbers, with each number being the sum of the two numbers directly above it, starting with the number 1 at the apex. The binomial coefficients in the expansion of \((a+b)^n\) are the same as the elements in the n-th row of Pascal's Triangle, beginning with the 0-th row corresponding to the \(n=0\) case. The elements of Pascal's Triangle can be calculated using the formula: \[C(n, k) = \frac{n!}{k!(n-k)!}\] This is the same formula used for calculating the binomial coefficients in the binomial expansion of \((a+b)^n\).

Now we will calculate the rows of Pascal's Triangle corresponding to the coefficients of each term in the given polynomial expressions. 0. Row 0 (n=0): \[C(0, 0) = 1\] 1. Row 1 (n=1): \[C(1, 0) = 1, C(1, 1) = 1\] 2. Row 2 (n=2): \[C(2, 0) = 1, C(2, 1)=2, C(2, 2)=1\] 3. Row 3 (n=3): \[C(3, 0)=1, C(3, 1)=3, C(3, 2)=3, C(3, 3)=1\] 4. Row 4 (n=4): \[C(4, 0)=1, C(4, 1)=4, C(4, 2)=6, C(4, 3)=4, C(4, 4)=1\] 5. Row 5 (n=5): \[C(5, 0)=1, C(5, 1)=5, C(5, 2)=10, C(5, 3)=10, C(5, 4)=5, C(5, 5)=1\] 6. Row 6 (n=6): \[C(6, 0)=1, C(6, 1)=6, C(6, 2)=15, C(6, 3)=20, C(6, 4)=15, C(6, 5)=6, C(6, 6)=1\] 7. Row 7 (n=7): \[C(7, 0)=1, C(7, 1)=7, C(7, 2)=21, C(7, 3)=35, C(7, 4)=35, C(7, 5)=21, C(7, 6)=7, C(7, 7)=1\] 8. Row 8 (n=8): \[C(8, 0)=1, C(8, 1)=8, C(8, 2)=28, C(8, 3)=56, C(8, 4)=70, C(8, 5)=56, C(8, 6)=28, C(8,7)=8, C(8,8)=1\] 9. Row 9 (n=9): \[C(9, 0)=1, C(9, 1)=9, C(9, 2)=36, C(9, 3)=84, C(9, 4)=126, C(9, 5)=126, C(9, 6)=84, C(9,7)=36, C(9,8)=9, C(9,9)=1\] The coefficients of the binomial expansion \((a+b)^0, (a+b)^1, \ldots , (a+b)^9\) can now be read directly from the rows of Pascal's Triangle listed above.