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Using all the letters of the word ARRANGEMENT how many type of various words utilizing all letters at a time have the right to be made such that both A, both E, both R both N happen together .

$egingroup$ In basic if you have $n$ objects via $r_1$ objects of one sort, $r_2$ objects of another,...,and also $r_k$ objects of the $k$th type, they can be arranged in $$fracn!(r_1!)(r_2!)dots(r_k!)$$ ways. $endgroup$

"ARRANGEMENT" is an eleven-letter word.

If tbelow were no repeating letters, the answer would certainly ssuggest be $11!=39916800$.

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However before, given that there are repeating letters, we have to divide to remove the duplicates appropriately.Tright here are 2 As, 2 Rs, 2 Ns, 2 Es

Therefore, tright here are $frac11!2!cdot2!cdot2!cdot2!=2494800$ means of arranging it.

Words ARRANGEMENT has $11$ letters, not all of them distinctive. Imagine that they are created on little bit Scrabble squares. And expect we have actually $11$ consecutive slots right into which to put these squares.

Tbelow are $dbinom112$ means to **choose** the slots wbelow the 2 A"s will go. For **each** of these means, tright here are $dbinom92$ means to decide where the two R"s will go. For eextremely decision about the A"s and also R"s, tright here are $dbinom72$ means to decide wbelow the N"s will go. Similarly, tbelow are currently $dbinom52$ ways to decide wbelow the E"s will go. That leaves $3$ gaps, and $3$ singleton letters, which deserve to be arranged in $3!$ ways, for a total of $$inom112inom92inom72inom523!.$$

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