To solve this problem, we need to determine the number of ways to distribute 10 different mugs between Lucy and Monica such that both receive an odd number of mugs.

Since the total number of mugs is 10, the possible distributions where both receive an odd number are:

• Lucy receives 1 mug and Monica receives 9 mugs.
• Lucy receives 3 mugs and Monica receives 7 mugs.
• Lucy receives 5 mugs and Monica receives 5 mugs.
• Lucy receives 7 mugs and Monica receives 3 mugs.
• Lucy receives 9 mugs and Monica receives 1 mug.

For each distribution, we calculate the number of combinations:

\(\binom{10}{1} + \binom{10}{3} + \binom{10}{5} + \binom{10}{7} + \binom{10}{9}\)

Calculating each term:

• \(\binom{10}{1} = 10\)
• \(\binom{10}{3} = 120\)
• \(\binom{10}{5} = 252\)
• \(\binom{10}{7} = 120\)
• \(\binom{10}{9} = 10\)

Summing these values gives:

\(10 + 120 + 252 + 120 + 10 = 512\)

Thus, there are 512 different ways to distribute the mugs such that both Lucy and Monica receive an odd number of mugs.