How many 7-digit numbers containing the digits 1, 2, 4, 5, 6, 7, 8, and 9 can be created so that the digits 4 and 6 appear together?
15640
17640
19440
16940
There are two ways to arrange the digits 4 and 5 next to each other: we can either have 45 or 54.
For each arrangement of the digits 4 and 5, we can treat them as a single digit and then arrange the remaining digits.
Consider the numbers 4 and 5 to be single digits. This results in a set of 7 numbers: 1, 2, 45, 6, 7, 8, and 9.
The number can start with any of the 7 digits, so we have 7 choices for the first digit.
For the second digit, we again have 7 choices, since we can use any of the 7 remaining digits.
For the third digit, we have only 1 choice, which is the combined digit.
For the fourth digit, we have 6 choices, since we cannot use the digit that was used for the second digit.
For the fifth digit, we have 5 choices, since we cannot use either of the digits that were used for the second or fourth digit.
For the sixth digit, we have 4 choices, since we cannot use either of the digits that were used for the second, fourth, or fifth digit.
For the seventh digit, we have 3 choices, since we cannot use either of the digits that were used for the second, fourth, fifth, or sixth digit.
Thus, the total number of 5-digit numbers that can be formed using the digits 1, 2, 4, 5, 6, 7, 8, and 9 such that the digits 4 and 5 appear together in the number is given by,
Therefore, the total number of ways the 7-digit number can be formed is 17640 ways.
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