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# How many ways to arrange?

+1
853
4
+816

How many ways are there to arrange the word, "Calculator"

May 14, 2018

#1
+4613
+2

Calculator: 10 letters

(a) repeats twice

(c) repeats twice

(l) repeats twice

So, $$\frac{10!}{2!*2!*2!}=453600$$

May 14, 2018
#2
+198
+2

Yes, like as what tertre said:

Here's a similar problem, How many ways are there to arrange the word, "Waterfall"

First, count how many letters, and see how many letters in the word repeat, then divide.

May 14, 2018
#3
+2442
+2

If the word did not have any repeating letters, then this problem would be much simpler.

For example, let's think of the number of possible arrangements for the not-so-arbitrary word "Melody."

• For the first letter, there are 6 total choices
• For the second letter, there are now 5 possible choices
• For the third letter, there are now 4 possible choices
• This goes on until the final letter where there is only one option.

Therefore, $$6*5*4*3*2*1$$ is the total number of combinations. In shorthand, this can be mathematically expressed with a factorial symbol: $$6!$$

If there are repetitions in the word, then simply divide by the number of arrangements of the repeated letters.

Therefore, in "calculator," there are three repeated letters. There are:

• 2 "c's"
• 2 "a's"
• 2 "l's"
 $$\frac{10!}{2!*2!*2!}$$ Now, just evaluate this. $$453600\text{ arrangements}$$
May 14, 2018
#4
+26381
+1

How many ways are there to arrange the word, "Calculator"

Count the letters in the word "Calculator":

$$\begin{array}{|c|r|} \hline \text{letter} & \text{count} \\ \hline c & 2 \\ a & 2 \\ l & 2 \\ u & 1 \\ t & 1 \\ o & 1 \\ r & 1 \\ \hline \text{sum} & \color{red}10 \\ \hline \end{array}$$

How many ways are there to arrange the word, "Calculator":

$$\begin{array}{|rcll|} \hline && \dfrac{ {\color{red}10} !}{2!2!2!1!1!1!1!} & | \quad 1! = 1 \\\\ &=& \dfrac{ {\color{red}10} !}{2!2!2!} & | \quad 2!=2 \\\\ &=& \dfrac{ {\color{red}10} !}{2\cdot 2\cdot 2} \\\\ &=& \dfrac{ {\color{red}10} !}{8} & | \quad 10! = 2\cdot 3\cdot4\cdot5\cdot6\cdot7\cdot8\cdot9\cdot10 \\\\ &=& \dfrac{ {2\cdot 3\cdot4\cdot5\cdot6\cdot7\cdot8\cdot9\cdot10} }{8} \\\\ &=& 2\cdot 3\cdot4\cdot5\cdot6\cdot7\cdot9\cdot10 \\\\ &=& 453600 \\ \hline \end{array}$$

May 15, 2018