36. The English alphabet has 5 vowels and 21 consonants. How many words with two different vowels and 2 different consonants can be formed from the alphabet?

36. In an English word there are 5 vowels and 21 consonants.

The number of ways of selecting 2 vowel out of 5 = ⁵C₂

= $\frac{5!}{2!\left(5?2\right)!}$

= 5 * 2 = 10

The number of ways of selecting 2 consonants out of 21 = ²¹C₂

= $\frac{21!}{2!\left(21?2\right)!}$

= 21 * 10

= 210

Therefore, the number of combinations of 2 vowels and 2 consonants is 10 * 210 = 2100

Each of these 2100 combinations has 4 letters which can be rearranged among themselves in 4! Ways.

Therefore, the required number of ways

= 4! * 2100

= 4 * 3 * 2 * 1 * 2100

= 50400

<p><strong>36.</strong> In an English word there are 5 vowels and 21 consonants.</p><p>The number of ways of selecting 2 vowel out of 5 = ⁵C₂</p><p>=&nbsp;<span title="Click to copy mathml"><math><mrow><mfrac><mrow><mn>5</mn><mo>!</mo></mrow><mrow><mn>2</mn><mo>!</mo><mrow><mo> (</mo><mrow><mn>5</mn><mo>? </mo><mn>2</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/1747823160php6Ld4T3.jpeg" alt="" width="103" height="52" loading="lazy"></picture></div></div><p>= 5 * 2 = 10</p><p>The number of ways of selecting 2 consonants out of 21 = ²¹C₂</p><p>=&nbsp;<span title="Click to copy mathml"><math><mrow><mfrac><mrow><mn>2</mn><mn>1</mn><mo>!</mo></mrow><mrow><mn>2</mn><mo>!</mo><mrow><mo> (</mo><mrow><mn>2</mn><mn>1</mn><mo>? </mo><mn>2</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/1747823203phpXAcGUQ.jpeg" alt="" width="107" height="45" loading="lazy"></picture></div></div><p>= 21 * 10</p><p>= 210</p><p>Therefore, the number of combinations of 2 vowels and 2 consonants is 10 * 210 = 2100</p><p>Each of these 2100 combinations has 4 letters which can be rearranged among themselves in 4! Ways.</p><p>Therefore, the required number of ways</p><p>= 4! * 2100</p><p>= 4 * 3 * 2 * 1 * 2100</p><p>= 50400</p>

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