Final answer:
The DJ can choose the first six songs from a playlist of 24 songs in 44,598,144 ways using the permutation formula.
Step-by-step explanation:
The DJ can choose the first six songs from a playlist of 24 songs in 24P6 ways, which is calculated using the permutation formula. The permutation formula is nPr = n! / (n - r)!, where n is the total number of items and r is the number of items chosen. In this case, n = 24 and r = 6.
Substituting the values into the formula: 24P6 = 24! / (24 - 6)!
Simplifying: 24P6 = 24! / 18!
Cancelling out common factors in the numerator and denominator: 24P6 = 24 × 23 × 22 × 21 × 20 × 19 = 44,598,144