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Problem 1058

Calculate the number of permutations of the letters $$a, b, c, d$$ taken two at a time.

Expert verified
There are 12 possible permutations of the letters $$a, b, c,$$ and $$d$$ taken two at a time.
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Step 1: Identify values of n and r

In this exercise, we have the letters $$a, b, c, d$$. So, the total number of objects (letters) is $$n = 4$$. We want to find permutations taken two at a time, so $$r = 2$$.

Step 2: Find Factorials

Now, we need the factorials of $$n=4$$ and $$(n-r) = (4-2) = 2$$. 1. For $$4!$$, we have: $$4! = 4\times 3\times 2\times 1 = 24$$ 2. For $$2!$$, we have: $$2! = 2\times 1 = 2$$

Step 3: Apply the Permutation Formula

Now, we will apply the formula to find the number of permutations: $$P(n, r) = \frac{n!}{(n-r)!}$$ Plugging in the values, we get: $$P(4, 2) = \frac{4!}{2!} = \frac{24}{2} = 12$$ So there are 12 possible permutations of the letters $$a, b, c,$$ and $$d$$ taken two at a time.

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