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# help

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A: Mack the bug starts at $(0,0)$ at noon and each minute moves one unit right or one unit up. He is trying to get to the point $(5,7)$. However, at $(2,3)$ there is a spider that will eat him if he goes through that point. In how many ways can Mack reach $(5,7)$?

B: How many ways are there to put 5 balls in 3 boxes if the balls are distinguishable but the boxes are not?

C: In how many ways can I place 5 different beads on a bracelet if rotations and flips of the bracelet are not treated as different arrangements?

Jun 28, 2019

#1
+23071
+1

A:

Mack the bug starts at $(0,0)$ at noon and each minute moves one unit right or one unit up. He is trying to get to the point $(5,7)$.

However, at $(2,3)$ there is a spider that will eat him if he goes through that point. In how many ways can Mack reach $(5,7)$?

Jun 28, 2019
#2
+23071
+1

B:
How many ways are there to put 5 balls in 3 boxes if the balls are distinguishable but the boxes are not?

I assume:
$$\text{ Let k=5 distinct balls } \\ \text{ Let n=3 identical boxes }$$

Formula:
$$\begin{array}{|rcll|} \hline \text{Distribution of k distinct Balls into n identical Boxes} =\sum \limits_{i=1}^{n}S(k,i) \\ \text{S(k,n), Stirling number of the second kind } \\ \hline \end{array}$$

$$S(k,n)$$:

$$\begin{array}{|rcll|} \hline \text{Distribution of 5 distinct Balls into 3 identical Boxes} &=&\sum \limits_{i=1}^{3}S(5,i) \\ &=& S(5,1)+ S(5,2)+S(5,3) \\ &=& 1 + 15 + 25 \\ &=& 41 \\ \hline \end{array}$$

Jun 28, 2019
edited by heureka  Jun 28, 2019