You have been assigned the task of designing new license plates for the state of california. You decide to use 7 spaces. The first three spaces must be a number, (repeats ok), the middle space must be a picture, there are 20 to choose from, and the last three spaces must be letters, (repeats ok), How many different license plates can the state produce? Please need help soon, your help is much appreciated.

Question

kropot72 · Answer

The number of different permutations of n different things taken k at a time with repetitions is
$$n ^{k}$$
Can you find the number of permutations of 26 letters taken 3 at a time with repetitions?

anonymous · Answer

so 26 npr 3?

kropot72 · Answer

Not really. If you look again at the formula for permutations with repetitions youo will see the number of permutations of 26 letters taken 3 at a time with repetitions where n = 26 and r =3 is given by
$$n ^{r}=26^{3}$$
Each of the 20 choices of picture can be taken with each of the combinations of letters. Therefore the number of arrangements of pictures and letters is given by
$$20\times 26^{3}$$
Can you now find the number of permutations of 10 digits taken 3 at a time with repetitions by plugging in the values into the formula for permutations with repetitions:
$$n ^{r}=\ ?$$

anonymous · Answer

so it would be 10^3

kropot72 · Answer

Good work! So the total number of different licence plates is given by:
$$10^{3}\times 20\times 26^{3}=you\ can\ calculate$$