The numbers of elements in the set $\left\{n\in \left\{1,2,3,....100\right\}|{\left(11\right)}^n>{\left(10\right)}^n+{\left(9\right)}^n\right\}$  is………….

  $11^n-9^n>10^n$             

$\Rightarrow {\left(10+1\right)}^n-{\left(10-1\right)}^n>10^n$           

->For  $n\ge 5$ the inequality Satisfies.

For n= 4.             2 [4 × 10³ + 4 × 10] > 10⁴

->8 × 1010 > 10⁴ which is a contradiction

$\therefore n\ge 5$ all values satisfies. Hence possible values of n is 96

<p>&lt;!- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1816421570"> </o:OLEObject>&lt;/xml&gt;&lt;! [endif]-&gt;</p><p>&nbsp;<span contenteditable="false"> <math> <mrow> <mn>1</mn> <msup> <mrow> <mn>1</mn> </mrow> <mrow> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mn>9</mn> </mrow> <mrow> <mi>n</mi> </mrow> </msup> <mo>&gt;</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mi>n</mi> </mrow> </msup> </mrow> </math> </span> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&lt;!- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1816421571"> </o:OLEObject>&lt;/xml&gt;&lt;! [endif]-&gt;</p><p><span contenteditable="false"> <math> <mrow> <mo>⇒</mo> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mn>1</mn> <mn>0</mn> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mn>1</mn> <mn>0</mn> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>n</mi> </mrow> </msup> <mo>&gt;</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mi>n</mi> </mrow> </msup> </mrow> </math> </span> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&lt;!- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1816421572"> </o:OLEObject>&lt;/xml&gt;&lt;! [endif]-&gt;</p><div><div><picture><source srcset="" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1754895533phpkcVZ5o.jpeg" alt="" width="369" height="39" loading="lazy"></picture></div></div><p>-&gt;For&nbsp;<span contenteditable="false"> <math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>5</mn> <mtext> </mtext> <mtext> </mtext> </mrow> </math> </span>the inequality Satisfies.</p><p>For n= 4. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 2 [4 × 10³ + 4 × 10] &gt; 10⁴</p><p>-&gt;8 × 1010 &gt; 10⁴ which is a contradiction</p><p><span contenteditable="false"> <math> <mrow> <mo>∴</mo> <mi>n</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math> </span>all values satisfies. Hence possible values of n is 96</p>

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