Short Quiz

Assertion: Every term in the expansion of \[ \left(x+\frac{1}{x}\right)^6 \] contains a non-negative integral power of \(x\).
Reason: The power of \(x\) in the \((r+1)\)th term is \(6-2r\).

Assertion (A): The expansion of \((a+b)^4\) requires calculating \({}^4C_0\), \({}^4C_1\), \({}^4C_2\), \({}^4C_3\), and \({}^4C_4\), which map directly to the numbers 1, 4, 6, 4, 1.
Reason (R): The addition of elements in Pascal's triangle row for index \(n\) generates the row for index \(n+1\) using the property \({}^nC_r + {}^nC_{r-1} = {}^{n+1}C_r\).

Assertion: For every positive integer \(n\), \[ {}^{n}C_0-{}^{n}C_1+{}^{n}C_2-\cdots+(-1)^n{}^{n}C_n=0. \]
Reason: Substituting \(x=-1\) in the expansion of \((1+x)^n\) gives \((1-1)^n=0\).

Assertion: \[ 98^5<100^5. \]
Reason: In the binomial expansion of \((100-2)^5\), every term after \(100^5\) is negative.