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First give everybody three pens, using 75 of them. Selection with Repetition The permutation and combination question we have done so far are basically about selecting objects. Something in particular that was missing: How many distinct values can be represented with 5 decimal digits? Here we are selecting items digits where repetition is allowed: We can actually answer this with just the product rule: Same as other combinations: Same as permutations with repetition: You walk into a candy store and have enough money for 6 pieces of candy.
The store has chocolate C , gummies G , and horrible Chinese candy H. How many different selections can you make? Here are some possible selections you might make: C then G then H. We don't want our candy to mix: How many solutions does this equation have in the non-negative integers? We select of them with repetition and that gives us a solution to the equation. The basic approach is: How do we deal with the repeated P? Let's pretend they're different: Then in the remaining 3 positions, permute the remaining 3 elements.
How many ways for a game that is played with two decks shuffled together? There are 11 letters, but four Is, four Ss, two Ps. More Counting Examples Your mother-in-law buys small gifts to give to relatives for Christmas, for reasons you don't understand. Each of the things is different, because she spends too much time shopping. There are 25 relatives to give gifts to.
How many ways are there to distribute the gifts? There's no fairness in this family In the above scenario, how many ways can the gifts be distributed so each person gets 40 items?
The next year, you are sent to the basement to arrange the gifts into 25 piles, with one for each person. You don't know who will be getting each pile, so the order of the piles doesn't matter.
How many arrangements are there for you to choose from? The next year, your mother-in-law buys identical pens to give to the family. You start to wonder what is going on. How many ways can they be distributed to 25 family members? How many ways if each person must get at least 3 pens? How many ways if each person gets 40 pens?
Solutions This is a permutation with repetition. For each gift, it can be given to one of 25 family members. First gift in 25 ways, second gift in 25 ways, …. Think of giving each family member 40 tokens with their name on it. Line the different gifts up in a row and have each family member drop their 40 tokens on gifts.
To distribute the gifts, we must select people to get each one. Order doesn't matter here, since the pens are identical. This is a combination with repetition problem: How many of those are there? We should first decide on a way to order permutations. Then we can generate them in that order, and be sure we found them all.
We will order them based on lexicographic order. Two permutations are equal if all elements are the same. The following algorithm will generate all permutations of elements of a set, in lexicographic order: Easy enough to prove by induction. To generate all combinations of a set we make this observation: The following algorithm will generate all combinations of elements of a set: An example of recursion making things easier: Which is more clear?
Which one do you believe is correct? I have had to do that zero times in my life. I have had to generate all of them.