+4 Hi I have a question
In how many ways can we distribute 13 pieces of identical candy to 5 kids, if the two youngest kids are twins and insist on receiving an equal number of pieces?
Thanks!
+1043 There are two main cases to consider when distributing the candy:
Case 1: The twins get the same amount of candy (more than 0 and less than all the candy)
Distribute identical candy to the twins: We can choose how many candies each twin gets in 11 ways (from 1 candy each to 6 candies each).
Distribute the remaining candy to the other 3 children: There are 13 - (2 * number of candies for twins) candies remaining.
We can distribute these candies among the 3 children in ways, following the stars and bars method (arranging 3 identical stars and (number of candies remaining - 1) non-identical bars).
Case 2: The twins each get 0 candies
Distribute all 13 candies to the other 3 children: We can distribute these candies in ways, following the stars and bars method (arranging 3 identical stars and (number of candies remaining - 1) non-identical bars).
Adding the number of ways for both cases gives us the total number of distributions.
Calculating the total ways
Case 1 ways:
Number of candy choices for twins (11 ways)
Ways to distribute remaining candy (dependent on the number chosen for twins)
Case 2 ways:
Ways to distribute candy to 3 children (without twins getting any)
For Case 1, we need to sum the ways to distribute the remaining candy for each candy choice for the twins (from 1 to 6). However, calculating this for each case can be cumbersome.
Simplifying Case 1 calculations
Notice that the ways to distribute the remaining candy only depends on the total number of candies remaining after giving the twins their share (and not on how many candies each twin got).
So, we can calculate the ways to distribute the candy for each total number of remaining candies (from 7 to 1) and reuse those values for all candy choices in Case 1 that result in the same total number of remaining candies.
Calculating reusable distribution ways
There are 3 ways to distribute 1 candy (1 child gets it, 2 children get it, or all 3 get 1 each).
There are 6 ways to distribute 2 candies (1 child gets 2, another gets none, or 2 children get 1 each).
There are 10 ways to distribute 3 candies (1 child gets 3, another gets none, 2 children get 1 each, or all 3 get 1 each).
We can continue using the stars and bars method for higher numbers of candies.
Total number of distributions
Case 1 ways:
We reuse the calculated ways to distribute remaining candy based on the total number remaining (from 7 ways for 1 candy remaining to 10 ways for 3 candies remaining).
This gives us a total of (3 + 6 + 10) * 11 = 198 ways (11 ways for choosing candy for twins multiplied by the sum of ways to distribute remaining candy (from 1 to 3 candies remaining for the twins).
Case 2 ways:
We can distribute candy to 3 children in 455 ways (using stars and bars for 13 candies and 2 dividers).
Total ways = Case 1 ways + Case 2 ways = 198 ways + 455 ways = 653 ways
Therefore, there are 653 ways to distribute 13 identical pieces of candy to 5 children if the two youngest (who are twins) receive the same amount of candy (and each can get between 0 and 6 candies).
