Distinguishable objects into distinguishable boxes where number of balls in each box varies 5 distinguishable balls distinguishable boxes where each box contains at least 2 balls, Distinguishable objects in distinguishable boxes so that there are k i objects in the i-th box : same as permutations with repetition. Indistinguishable objects in distinguishable boxes : stars and bars again. Indistinguishable objects in indistinguishable boxes : partitions… Stars and Bars 1.
Distinct objects in indistinguishable boxes When placing k distinguishable objects into n indistinguishable boxes , what matters?! Each object needs to be in some box .! No object is in two boxes . We have rediscovered . So ask “How many set partitions are there of a set with k objects ?” Or even, “How many set partitions are there of k objects into n parts?”, Case2: INDISTINGUISHABLE OBJECTS AND DISTINGUISHABLE BOXES Counting the number of ways of placing n indistinguishable objects into k distinguishable boxes turns out to be the same as counting the number of n-combinations for a set with k elements when repeti tions are allowed.
Case2: INDISTINGUISHABLE OBJECTS AND DISTINGUISHABLE BOXES Counting the number of ways of placing n indistinguishable objects into k distinguishable boxes turns out to be the same as counting the number of n-combinations for a set with k elements when repeti tions are allowed.
: Use the previous and the addition principle on the cases: rballs in 1 box none empty, rballs into 2 boxes none empty, etc. n! r n: Put the balls into indistinguishable boxes (r n ways). The boxes are now distinguishable by their contents. Then put labels on the boxes (n! ways). Later we show by inclusion-exclusion that: n! ˆ r n ? = Xn i=0 …
Indistinguishable to distinguishable (Balls and Urns / Sticks and Stones / Stars and Bars) This is the Balls and Urns technique. In general, if one has indistinguishable objects that one wants to distribute to distinguishable containers, then there are ways to do so.
It is used to solve problems of the form: how many ways can one distribute indistinguishable objects into distinguishable bins? We can imagine this as finding the number of ways to drop balls into urns, or equivalently to arrange balls and dividers. For example, represent the ways to put objects in bins. The number of ways to do such is , or .
Distinguishable Balls Binomial Theorem Onto Functions and the Stirling Numbers of Second Kind Indistinguishable Balls and Distinguishable Boxes Indistinguishable Balls in Indistinguishable Boxes Lattice Paths and Catalan Numbers Catalan Numbers Continued Generalizations Advanced Counting and Generating Functions. Pigeonhole Principle, Each box must contain exactly one ball (=1) n k (formula 1) n P k (formula 2) S(k,n) × n! (formula 3) n P n if k=n 0 if k?n (formula 4) (Note: n P n =n!) Distribution of k identical balls into n distinct boxes : No Restrictions: Each box can contain at most one ball (? 1) Each box must contain at least one ball (? 1) Each box must contain exactly one ball (=1)
