How many 4-digit numbers can be formed using the numbers 5, 6, 7 and 8? How many of the 4-digit numbers are less than 7000?
Since there are 4 choices for each digit, there are 4^4 = 256 different numbers.
Only half of them begin with 5 or 6
24
To find the number of 4-digit numbers that can be formed using the numbers 5, 6, 7, and 8, we need to consider the position of each digit.
For the first digit (thousands place), we have four options (5, 6, 7, and 8) since the number cannot start with zero.
For the second digit (hundreds place), we have three options. Since one of the digits has already been used for the thousands place, we are left with the remaining three.
For the third digit (tens place), we also have three options. Again, one of the digits has already been used for the thousands place, and another digit was used for the hundreds place.
Finally, for the last digit (ones place), we have two options left (since two digits were used for the thousands, hundreds, and tens places).
To find the total number of 4-digit numbers, we multiply the number of options for each position:
4 options for the thousands place × 3 options for the hundreds place × 3 options for the tens place × 2 options for the ones place = 72.
Therefore, there are 72 different 4-digit numbers that can be formed using the numbers 5, 6, 7, and 8.
To determine how many of these 4-digit numbers are less than 7000, we need to focus on the thousands place. Since the thousands place can only be one of the digits 5 or 6 (as numbers less than 7000 cannot start with 7 or 8), there are two options for the thousands place.
For the hundreds, tens, and ones places, we can still choose from the remaining three digits (6, 7, and 8) since these positions are not constrained by the condition of being less than 7000.
So, the number of 4-digit numbers that are less than 7000 can be calculated as:
2 options for the thousands place × 3 options for the hundreds place × 3 options for the tens place × 2 options for the ones place = 36.
Therefore, there are 36 4-digit numbers that can be formed using the numbers 5, 6, 7, and 8, which are less than 7000.