How many four digit numbers use the digit 7 two or more times?
To determine the number of four-digit numbers that use the digit 7 two or more times, we can break down the problem into cases:
Case 1: Two 7's
In this case, we have two options for the placements of the two 7's. The remaining two digits can be any of the nine possible digits (0-9 excluding 7). Therefore, the number of four-digit numbers with two 7's is 2 * 9 * 9 = 162.
Case 2: Three 7's
Similar to the previous case, we have three possible placements for the three 7's. The remaining digit can be any of the nine possible digits. Thus, the number of four-digit numbers with three 7's is 3 * 9 = 27.
Case 3: Four 7's
In this case, there is only one possible arrangement: all four digits are 7's. Hence, there is only 1 four-digit number with four 7's.
To get the total number of four-digit numbers that use the digit 7 two or more times, we sum up the results from the three cases: 162 + 27 + 1 = 190.
Therefore, there are 190 four-digit numbers that use the digit 7 two or more times.