5 a) How many even numbers can be formed from the digits 1, 2, 3, 4, 5?
b) How many of these numbers are greater than 3000?
can someone answer part a
A motorcycle licence plate consists of two or three letters followed by four digits. How many licence plates can be made?
To find the number of even numbers that can be formed from the digits 1, 2, 3, 4, and 5, we need to consider the following:
a) How many choices do we have for the first digit?
Since we want even numbers, the first digit cannot be 1 or 3. It can only be 2, 4, or 5. So we have 3 choices for the first digit.
b) How many choices do we have for the second, third, fourth, and fifth digits?
Since we can repeat digits, we have all 5 choices for each of these digits.
To find the total number of even numbers, we multiply the number of choices for each digit:
a) Number of even numbers = number of choices for the first digit × number of choices for the second digit × number of choices for the third digit × number of choices for the fourth digit × number of choices for the fifth digit
= 3 × 5 × 5 × 5 × 5
= 3 × 5^4
= 3 × 625
= 1875
So there are 1875 even numbers that can be formed from the digits 1, 2, 3, 4, and 5.
Now, to find the number of even numbers greater than 3000, we need to consider the following:
a) How many choices do we have for the first digit?
Since we want the number to be greater than 3000, the first digit can only be 4 or 5. So we have 2 choices for the first digit.
b) How many choices do we have for the second, third, and fourth digits?
Since we can repeat digits, we have all 5 choices for each of these digits.
To find the total number of even numbers greater than 3000, we multiply the number of choices for each digit:
b) Number of even numbers greater than 3000 = number of choices for the first digit × number of choices for the second digit × number of choices for the third digit × number of choices for the fourth digit
= 2 × 5 × 5 × 5
= 2 × 5^3
= 2 × 125
= 250
So there are 250 even numbers that can be formed from the digits 1, 2, 3, 4, and 5, which are greater than 3000.
You have conflicting restrictions, so take it in cases
To be greater than 3000, the number must start with either 4 or 5
case1 : it start with a 4, that means you have 1 choice to make it even, namely the 2
number of ways = 1*3*2*1*1 = 6
case2 : it starts with a 5, so you have 2 choices for the unit digit
number of ways = 1*3*2*1*2 = 12
So you have 18 such numbers