Question

which of the following statements regarding pascals triangle are correct?
a. each number not at the end of a row, is the sum of the numbers that are directly to its right and left, in the row below.
b. the nth row gives the coefficients in the expansion of (x + y)^n.
c. the nth row gives the coefficients in the expansion of (x + y)^{n - 1}.
d. the first and last numbers of a row are always 1.

Explanation:

• Option A: In Pascal's triangle, each number not at the end of a row is the sum of the two numbers directly above it (in the row above), not below. So, this statement is incorrect.
• Option B: The binomial theorem states that \((x + y)^n=\sum_{k = 0}^{n}\binom{n}{k}x^{n - k}y^{k}\), and the coefficients \(\binom{n}{k}\) for \(k=0,1,\cdots,n\) are given by the \((n + 1)\) - th row of Pascal's triangle. So, this statement is incorrect.
• Option C: By the binomial theorem \((x + y)^{n-1}=\sum_{k=0}^{n - 1}\binom{n-1}{k}x^{(n - 1)-k}y^{k}\), and the coefficients \(\binom{n-1}{k}\) for \(k = 0,1,\cdots,n-1\) are given by the \(n\) - th row of Pascal's triangle. This statement is correct.
• Option D: The first element of the \(n\) - th row of Pascal's triangle is \(\binom{n-1}{0}=1\) and the last element is \(\binom{n - 1}{n-1}=1\). So, this statement is correct.

Answer:

C. The \(n\)th row gives the coefficients in the expansion of \((x + y)^{n - 1}\); D. The first and last numbers of a row are always \(1\)