To solve this problem, we need to find the number of ways to distribute 12 different cakes between Alex and James such that both receive an odd number of cakes.
The possible distributions where both receive an odd number of cakes are:
- Alex receives 1 cake and James receives 11 cakes.
- Alex receives 3 cakes and James receives 9 cakes.
- Alex receives 5 cakes and James receives 7 cakes.
- Alex receives 7 cakes and James receives 5 cakes.
- Alex receives 9 cakes and James receives 3 cakes.
- Alex receives 11 cakes and James receives 1 cake.
For each distribution, we calculate the number of ways to choose the cakes for Alex, and the remaining cakes automatically go to James. This is given by the combination formula:
\(\binom{12}{1} + \binom{12}{3} + \binom{12}{5} + \binom{12}{7} + \binom{12}{9} + \binom{12}{11}\)
Calculating each term:
- \(\binom{12}{1} = 12\)
- \(\binom{12}{3} = 220\)
- \(\binom{12}{5} = 792\)
- \(\binom{12}{7} = 792\)
- \(\binom{12}{9} = 220\)
- \(\binom{12}{11} = 12\)
Summing these values gives:
\(12 + 220 + 792 + 792 + 220 + 12 = 2048\)
Therefore, the number of ways to distribute the cakes such that both receive an odd number is 2048.
