Solution:

Given that the numbers a, b, c, d, and e form an arithmetic progression (A.P.), we can write the general form of the A.P. as:

a, a + d, a + 2d, a + 3d, a + 4d

Where 'a' is the first term and 'd' is the common difference.

To find the value of a-4b+6c-4d+e, we substitute the values of the terms in the A.P. into the expression:

a - 4b + 6c - 4d + e

Let's solve it step by step:

Step 1: Finding the common difference (d)
Since the numbers form an A.P., the difference between consecutive terms is constant.
So, we can find the common difference (d) by subtracting any two consecutive terms.
Let's subtract the second term (a + d) from the first term (a):

(a + d) - a = d

Therefore, the common difference (d) is equal to d.

Step 2: Substituting the values of the terms
Now, substitute the values of the terms in the expression:

a - 4b + 6c - 4d + e
= a - 4(a + d) + 6(a + 2d) - 4(a + 3d) + (a + 4d)
= a - 4a - 4d + 6a + 12d - 4a - 12d + a + 4d
= (a - 4a - 4a + a) + (-4d + 12d - 12d + 4d)
= -6a - 4d

Therefore, the value of a - 4b + 6c - 4d + e is -6a - 4d.

Conclusion:
The value of a - 4b + 6c - 4d + e for the given arithmetic progression is -6a - 4d.