In how many ways can we arrange the alphabets of the word ARRANGE such that all the vowels are together?

<p>Consider the vowels A, E as one unit. Besides this, we have 2R, 1N, 1G that is a total of 5 things out of which the 2 R are identical.</p><p>Required number of ways&nbsp;<math><mrow><mo>=</mo><mfrac><mrow><mn>5</mn><mo>!</mo></mrow><mrow><mn>2</mn><mo>!</mo></mrow></mfrac></mrow></math></p><p>Now A, A and E can be arranged within themselves in ways.&nbsp;<math><mrow><mfrac><mrow><mn>3</mn><mo>!</mo></mrow><mrow><mn>2</mn><mo>!</mo></mrow></mfrac></mrow></math></p><p>Required answer =&nbsp;<math><mrow><mfrac><mrow><mn>5</mn><mo>!</mo><mtext> </mtext><mo>×</mo><mn>3</mn><mo>!</mo></mrow><mrow><mn>2</mn><mo>!</mo><mtext> </mtext><mo>×</mo><mtext> </mtext><mn>2</mn><mo>!</mo></mrow></mfrac></mrow></math></p>

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