Anna was opening a new restaurant, so she went to the sign store to get letters to make a sign to hang above the storefront. When she got to the sign store, the only letters they had in stock were two copies of the letter 'X,' two copies of the letter 'E,' and one copy of the letter 'V.' Anna decided to buy all the letters and use them to make the name of her store. How many different names can she give her store using these 5 letters?
5!/(1! 2! 2!) = 30
To determine the number of different names Anna can give her store using these 5 letters ('X', 'X', 'E', 'E', 'V'), we need to calculate the number of permutations.
A permutation is an arrangement of objects in a specific order. In this case, we are arranging the letters to form different store names. Since two of the letters are repeated (the 'X' and 'E'), we need to take into account that these repeated letters are indistinguishable and can be swapped with each other without changing the name.
We can use the formulas for permutations with repetitions to solve this problem. The formula is:
P(n; r1, r2, ..., rk) = n! / (r1! * r2! * ... * rk!)
Where n is the total number of objects and r1, r2, ..., rk are the numbers of repetitions for each object.
In this case, we have n = 5 (total number of letters) and there are two 'X's and two 'E's. Hence, r1 = 2 (number of repetitions for 'X') and r2 = 2 (number of repetitions for 'E').
Plugging these values into the formula, we get:
P(5; 2, 2) = 5! / (2! * 2!)
Calculating this:
P(5; 2, 2) = (5 * 4 * 3 * 2 * 1) / ((2 * 1) * (2 * 1))
P(5; 2, 2) = 120 / (2 * 2)
P(5; 2, 2) = 120 / 4
P(5; 2, 2) = 30
Therefore, Anna can create 30 different names for her store using the available letters.