The number of positive integers k such that the constant term in the binomial expansion of ${\left(2x^3+\frac{3}{x^k}\right)}^{12},x\ne 0is{2}^8.\mathcal{l},where\mathcal{l}$  is an odd integer, is………..

<p><math><mrow><msup><mrow><mrow><mo> (</mo><mrow><mn>2</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mfrac><mrow><mn>3</mn></mrow><mrow><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mn>2</mn></mrow></msup></mrow></math></p><p>gen term = <picture><source srcset="" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1752484588phphO25Yl.jpeg" alt="" width="118" height="42" loading="lazy"></picture></p><p><math><mrow><msup><mrow><mo>=</mo></mrow><mrow><mn>1</mn><mn>2</mn></mrow></msup><msub><mrow><mi>C</mi></mrow><mrow><mi>r</mi></mrow></msub><msup><mrow><mn>2</mn></mrow><mrow><mn>1</mn><mn>2</mn><mo>−</mo><mi>r</mi></mrow></msup><mo>.</mo><msup><mrow><mn>3</mn></mrow><mrow><mi>r</mi></mrow></msup><mo>.</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn><mn>6</mn><mo>−</mo><mn>3</mn><mi>r</mi><mo>−</mo><mi>r</mi><mi>k</mi></mrow></msup></mrow></math></p><p>For constant term</p><p>36 – 3r – rk = 0</p><p><math><mrow><mo>⇒</mo><mi>k</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mn>6</mn><mo>−</mo><mn>3</mn><mi>r</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></math></p><p>for r = 1, 2, 4</p><p>¹²C_r2^12-r&gt;2⁸</p><p>Possible values of k = 3, 1</p>

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