<p>The number of distinct terms in the expansion of \(\left(x + \dfrac{1}{x} + x^2 + \dfrac{1}{x^2}\right)^{15}\) is/are (with respect to different power of \(x\))</p>
Step-by-Step Solution
Key Concept: Rewrite the expression as $(x + x^{-1} + x^2 + x^{-2})^{15}$ and expand using multinomial theorem. Each term has the form $x^{a+2b-c-2d}$ where $a+b+c+d=15$, so we need to find distinct values of the exponent $a+2b-c-2d$.
<p><strong>Step 1:</strong> Rewrite the expression as $(x + x^{-1} + x^2 + x^{-2})^{15}$</p><p><strong>Step 2:</strong> In the multinomial expansion, each term is $x^{a + 2b - c - 2d}$ where $a, b, c, d \geq 0$ and $a + b + c + d = 15$. Here $a$ is the power from $x$, $b$ from $x^2$, $c$ from $x^{-1}$, and $d$ from $x^{-2}$.</p><p><strong>Step 3:</strong> Let $n = a + 2b - c - 2d$ (the exponent of $x$). We need to find the range of possible values of $n$.</p><p><strong>Step 4:</strong> Maximum value: When $a = 15, b = 0, c = 0, d = 0$: $n_{max} = 15 + 0 - 0 - 0 = 15$</p><p><strong>Step 5:</strong> Minimum value: When $a = 0, b = 0, c = 0, d = 15$: $n_{min} = 0 + 0 - 0 - 30 = -30$</p><p><strong>Step 6:</strong> Since $n = a + 2b - c - 2d$ and $a + b + c + d = 15$, we can write $n = a + 2b - c - 2d = (a - c) + 2(b - d)$. For any integer $n$ with $-30 \leq n \leq 15$, we can verify that a valid non-negative integer solution exists by choosing appropriate values of $a, b, c, d$.</p><p><strong>Step 7:</strong> The number of distinct integer values from $-30$ to $15$ inclusive is $15 - (-30) + 1 = 46$</p><p>∴ Answer: <strong>46</strong></p>
Correct Answer: B