28. Determine the number of 5 card combinations out of a deck of 52 cards if thereis exactly one ace in each combination.

28. In a deck of 52 cards there are 4 ace cards. The required number of ways of selecting one ace card from the four = ⁴C₁ = $\frac{4!}{1!\left(4-1\right)!}$ = $\frac{4!}{3!}$ = $\frac{4\times 3!}{3!}$ = 4

After selecting one ace we need to select the remaining 4 card from the remaining 48 card to have a combination of 5 cards. The required number of ways

= ⁴⁸C₄

= $\frac{48!}{4!\left(48-4\right)!}$

= 1,94,580

Therefore, the total number of ways for selecting 5 card combination out of a deck of 52 cards if there is exactly one ace in each combination

= ⁴C₁×⁴⁸C₄

= 4 × 1,94,580

= 7,78,320

<p><strong>28.</strong> In a deck of 52 cards there are 4 ace cards. The required number of ways of selecting one ace card from the four = ⁴C₁ = <span title="Click to copy mathml"><math><mrow><mfrac><mrow><mn>4</mn><mo>!</mo></mrow><mrow><mn>1</mn><mo>!</mo><mrow><mo> (</mo><mrow><mn>4</mn><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>!</mo></mrow></mfrac></mrow></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><mrow><mfrac><mrow><mn>4</mn><mo>!</mo></mrow><mrow><mn>3</mn><mo>!</mo></mrow></mfrac></mrow></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><mrow><mfrac><mrow><mn>4</mn><mo>×</mo><mn>3</mn><mo>!</mo></mrow><mrow><mn>3</mn><mo>!</mo></mrow></mfrac></mrow></math></span>&nbsp;= 4</p><p>After selecting one ace we need to select the remaining 4 card from the remaining 48 card to have a combination of 5 cards. The required number of ways</p><p>= ⁴⁸C₄</p><p>=&nbsp;<span title="Click to copy mathml"><math><mrow><mfrac><mrow><mn>4</mn><mn>8</mn><mo>!</mo></mrow><mrow><mn>4</mn><mo>!</mo><mrow><mo> (</mo><mrow><mn>4</mn><mn>8</mn><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow><mo>!</mo></mrow></mfrac></mrow></math></span></p><div><div><picture><source srcset="" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1747819524phpppZZ9S.jpeg" alt="" width="185" height="94" loading="lazy"></picture></div></div><p>= 1,94,580</p><p>Therefore, the total number of ways for selecting 5 card combination out of a deck of 52 cards if there is exactly one ace in each combination</p><p>= ⁴C₁×⁴⁸C₄</p><p>= 4 × 1,94,580</p><p>= 7,78,320</p>

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