10 Question MCQ (including Assertion)

The simplified form of \((a+b)^4-(a-b)^4\) is

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: The coefficient of \(x^2\) in \((1+x)^5\) is \(10\).
Reason: Binomial coefficients satisfy \[ {}^{n}C_r={}^{n}C_{n-r}. \]

Assertion (A): For distinct integers \(a\) and \(b\), \((a-b)\) is a factor of \(a^n - b^n\) for every positive integer \(n\).
Reason (R): \(a^n\) can be written as \(((a-b) + b)^n\), and upon expansion using the Binomial Theorem, every term except the last one contains \((a-b)\) as a factor.