Introduction:
We are given an arithmetic progression (A.P) with the first term (a) as 24 and the common difference (d) as -3. We need to determine how many terms of this A.P must be taken in order to obtain a sum of 78.

Formula for the sum of an A.P:
The sum of the first n terms of an arithmetic progression (S_n) can be calculated using the formula:
S_n = (n/2) * [2a + (n-1)d]

Step 1: Substitute the given values:
Let's substitute the given values into the formula:
78 = (n/2) * [2(24) + (n-1)(-3)]

Step 2: Simplify the equation:
Let's simplify the equation by expanding and combining like terms:
78 = (n/2) * [48 - 3n + 3]
78 = (n/2) * [51 - 3n]

Step 3: Eliminate the fraction:
To eliminate the fraction, we can multiply both sides of the equation by 2:
2 * 78 = n * [51 - 3n]
156 = n * (51 - 3n)

Step 4: Rearrange the equation:
Let's rearrange the equation into a quadratic form:
156 = 51n - 3n^2
3n^2 - 51n + 156 = 0

Step 5: Solve the quadratic equation:
We can solve the quadratic equation by factoring or using the quadratic formula. However, in this case, the equation can be factored as follows:
(n - 6)(3n - 26) = 0

Setting each factor equal to zero gives us two possible solutions:
n - 6 = 0, which gives n = 6
3n - 26 = 0, which gives n = 26/3 = 8.67 (not a valid solution for the number of terms)

Step 6: Determine the number of terms:
From the solutions obtained, n = 6 is a valid solution since the number of terms in an arithmetic progression must be a positive whole number.

Conclusion:
In order to obtain a sum of 78 in the given arithmetic progression, we need to take 6 terms.