Question

question 9 of 10
which of the following statements regarding pascals triangle are correct?
a. the coefficients of $(x + y)^n$ are given by the $n$th row of pascals
triangle.
b. each row is a palindrome - that is, it is the same if read from
left to right or from right to left.
c. to generate pascals triangle, add adjacent terms on a row to
determine the term directly below them.
d. the $n$th row gives the coefficients in the expansion of $(x + y)^{n - 1}$.

Explanation:

• Option A: The coefficients of \((x + y)^n\) are given by the \((n + 1)\)th row of Pascal's triangle, not the \(n\)th row. So, this statement is incorrect.
• Option B: Each row of Pascal's triangle is a palindrome. For example, row \(n=3\) is \(1\ 3\ 3\ 1\). Reading from left - to - right or right - to - left gives the same sequence. This statement is correct.
• Option C: To generate Pascal's triangle, we use the rule \(C(n,k)=C(n - 1,k-1)+C(n - 1,k)\). That is, we add adjacent terms on a row to determine the term directly below them. This statement is correct.
• Option D: The \(n\)th row of Pascal's triangle gives the coefficients in the expansion of \((x + y)^{n - 1}\). For example, the \(1\)st row (\(1\)) gives the coefficients of \((x + y)^{0}=1\), the \(2\)nd row (\(1\ 1\)) gives the coefficients of \((x + y)^{1}=x + y\). This statement is correct.

Answer:

B. Each row is a 'palindrome' – that is, it is the same if read from left to right or from right to left.
C. To generate Pascal's triangle, add adjacent terms on a row to determine the term directly below them.
D. The \(n\)th row gives the coefficients in the expansion of \((x + y)^{n - 1}\).