Question

How many 5-number license plates can be made using the digits 1, 2, 3, 4, 5, 6, 7, if an odd digit must come first and
a. repetitions are allowed
b. repetitions are not allowed?

Answer 1

Answer: For Part A, there are 9,604 possibilities. For Part B, there are 1,440 possibilities.

To find the total number of possibilities, we will use the Fundamental Counting Principle. To do this, we find the number of possibilities for each position and then multiply them.

For Part B:
There are only 4 possibilities for the first position, since it must be odd (1, 3, 5, 7). The for the next 4 digits, all 7 numbers are a possibility.

4 x 7 x 7 x 7 x 7 = 9604

For Part B:
We again start with 4, however the numbers can be repeated. So we reduced the number of choice by 1 each time.
4 x 6 x 5 x 4 x 3 = 1440

Answer 2

The 5-digit number license plates can be made using the digits 1, 2, 3, 4, 5, 6, 7 and if an odd digit must come first with repetition and without repetition is 9604 and 1440 respectively.

What are permutation and combination?

A permutation is an act of arranging the objects or elements in order. Combinations are the way of selecting objects or elements from a group of objects or collections, in such a way the order of the objects does not matter.

5-number license plates can be made using the digits 1, 2, 3, 4, 5, 6, 7.

If an odd digit must come first.

A. With repetition.

$\rm Number \ of \ ways = 4 \times 7^4\\\\Number \ of \ ways = 4 \times 2401\\\\Number \ of \ ways = 9604$

B. Without repetition

$\rm Number \ of \ ways = 4 \times 6\times5\times4\times3\\\\Number \ of \ ways = 1440$

More about the permutation and the combination link is given below.

https://brainly.com/question/11732255