<p>Find the number of nonzero terms in the expansion of <span>\((1 + 3\sqrt{2}\,x)^9 + (1 - 3\sqrt{2}\,x)^9\)</span>.</p>
Step-by-Step Solution
Key Concept: When adding two binomial expansions with opposite middle terms, all odd-powered terms cancel out, leaving only even-powered terms. Recognize that $(1+a)^n + (1-a)^n$ contains only even powers of $a$.
<p><strong>Step 1:</strong> Apply binomial theorem to each term separately.</p><p>$(1 + 3\sqrt{2}x)^9 = \sum_{r=0}^{9} \binom{9}{r}(3\sqrt{2}x)^r$</p><p>$(1 - 3\sqrt{2}x)^9 = \sum_{r=0}^{9} \binom{9}{r}(-1)^r(3\sqrt{2}x)^r$</p><p><strong>Step 2:</strong> Add the two expansions.</p><p>When we add them, the term $\binom{9}{r}(3\sqrt{2}x)^r$ appears with coefficient $[1 + (-1)^r]$.</p><p><strong>Step 3:</strong> Analyze which terms survive.</p><p>$[1 + (-1)^r] = \begin{cases} 2 & \text{if } r \text{ is even} \\ 0 & \text{if } r \text{ is odd} \end{cases}$</p><p><strong>Step 4:</strong> Count nonzero terms.</p><p>Only even values of $r$ contribute: $r = 0, 2, 4, 6, 8$</p><p>This gives exactly 5 nonzero terms.</p><p>∴ Answer: <strong>5</strong></p>
Correct Answer: 5