Question

which of the following statements regarding pascals triangle are correct?
a. the nth row gives the coefficients in the expansion of (x + y)^n - 1.
b. the coefficients of (x + y)^n are given by the nth row of pascals triangle.
c. to generate pascals triangle, add adjacent terms on a row to determine the term directly below them.
d. each row is a palindrome - that is, it is the same if read from left to right or from right to left.

Explanation:

• Option A: The \(n\)th row of Pascal's triangle gives the coefficients in the expansion of \((x + y)^{n-1}\). For example, the first row (\(n = 1\)) is \(1\), and \((x + y)^{0}=1\).
• Option B: The coefficients of \((x + y)^{n}\) are given by the \((n + 1)\)th row of Pascal's triangle. For example, \((x + y)^{2}=x^{2}+2xy + y^{2}\), and the third row of Pascal's triangle is \(1,2,1\).
• Option C: To generate Pascal's triangle, we add adjacent terms on a row to determine the term directly below them. For example, in the second row \(1,1\), \(1+1 = 2\) which is the middle term of the third row \(1,2,1\).
• Option D: Each row of Pascal's triangle is a palindrome. For example, the fourth row \(1,3,3,1\) reads the same from left - to - right and right - to - left.

Answer:

C. To generate Pascal's triangle, add adjacent terms on a row to determine the term directly below them.
D. Each row is a "palindrome" - that is, it is the same if read from left to right or from right to left.