Solution:

Given: 4-digit numbers, each greater than 1000 and each having all four digits distinct, are there with 7 coming before 3.

To find: Number of such numbers

Let us first consider the number of such 4-digit numbers that can be formed using the digits 0 to 9, without any restrictions.

Number of ways of selecting the first digit = 9 (as the number cannot start with 0 or 1)

Number of ways of selecting the second digit = 9 (as any digit except the one already selected can be chosen)

Number of ways of selecting the third digit = 8 (as any digit except the two already selected can be chosen)

Number of ways of selecting the fourth digit = 7 (as any digit except the three already selected can be chosen)

Total number of such 4-digit numbers = 9 x 9 x 8 x 7 = 4536

Now, we need to subtract the number of 4-digit numbers that have 3 before 7.

Number of ways of selecting the first digit = 9 (as the number cannot start with 0 or 1)

Number of ways of selecting the second digit = 9 (as any digit except 3 can be chosen)

Number of ways of selecting the third digit = 1 (as 3 must be selected)

Number of ways of selecting the fourth digit = 7 (as any digit except the three already selected can be chosen)

Total number of such 4-digit numbers = 9 x 9 x 1 x 7 = 567

Therefore, the required number of 4-digit numbers = 4536 - 567 = 3969

Finally, we need to subtract the number of 4-digit numbers that have 3 before 7 and start with 1.

Number of ways of selecting the second digit = 9 (as any digit except 3 can be chosen)

Number of ways of selecting the third digit = 1 (as 3 must be selected)

Number of ways of selecting the fourth digit = 7 (as any digit except the three already selected can be chosen)

Total number of such 4-digit numbers = 9 x 1 x 7 = 63

Therefore, the required number of 4-digit numbers = 3969 - 63 = 3906

Finally, we need to subtract the number of 4-digit numbers that have 3 before 7 and start with 1 or 2.

Number of ways of selecting the second digit = 8 (as 1 or 2 cannot be chosen)

Number of ways of selecting the third digit = 1 (as 3 must be selected)

Number of ways of selecting the fourth digit = 7 (as any digit except the three already selected can be chosen)

Total number of such 4-digit numbers = 8 x 1 x 7 = 56

Therefore, the required number of 4-digit numbers = 3906 - 56 = 3850

Finally, we need to subtract the number of 4-digit numbers that have 3 before 7 and start with 1, 2, or 3.

Number of ways of selecting the third digit = 1 (as 3 must be selected)

Number of ways of selecting the fourth digit = 6 (as any digit except the three already selected can be chosen)

Total number of such 4