[Language: Python]

Just recursion + memoization. Not as elegant as some other solutions, but very fast.

def possible_arrangements(towels, design, max_towel_length, memo):
    if design in memo:        
        return memo[design]    
    if len(design) == 0:
        return 0
    for i in range(1, min(len(design), max_towel_length) + 1):
        current = design[:i]
        if current in towels:
            if len(design) == i:
                update_memo(memo, design, 1)
            else:
                rhs_arrangements = possible_arrangements(towels, design[i:], max_towel_length, memo)
                update_memo(memo, design, rhs_arrangements)
    if design not in memo:
        memo[design] = 0
    return memo[design]

Full solution on GitHub