Find the total number of ways to form a 4 digit number(without repetition) from the digits 0,1,2, ..., 6.6 + ⁶P₃⁷P₄5 x ⁵P₃6 x ⁶P₃

The problem can be divided into two cases:

Case 1: The 4-digit number does not start with 0. In this case, the first digit can be any of the 6 non-zero digits (1,2,3,4,5,6), and the remaining 3 digits can be any of the remaining 6 digits. So, the total number of ways is 6P1 * 6P3.

Case 2: The 4-digit number starts with 0. In this case, the first digit must be 0, and the remaining 3 digits can be any of the remaining 6 digits. So, the total number of ways is 1 * 6P3.

Adding the two cases together, the total number of ways to form a 4-digit number from the digits 0,1,2,...,6 without repetition is 6P1 * 6P3 + 1 * 6P3.

Now, let's calculate the values:

6P1 = 6 6P3 = 654 = 120 1 * 6P3 = 120

So, the total number of ways is 6*120 + 120 = 840.

For the second part of the question, we have to calculate the values of 6P3, 7P4, 5P3, and 6P3.

6P3 = 654 = 120 7P4 = 7654 = 840 5P3 = 543 = 60 6P3 = 65*4 = 120

So, the value of the expression 6P3 + 7P4 * 5P3 * 6P3 is 120 + 84060120 = 6051120.