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?
+26367 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 $(5,7)$?
see: https://web2.0calc.com/questions/help_26367#r5
or see: https://web2.0calc.com/questions/help_26367#r6
+26367 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:
\(\begin{array}{|rcll|} \hline \text{Distribution of $k$ distinct Balls into $n$ identical Boxes} =\sum \limits_{i=1}^{n}S(k,i) \\ \text{$S(k,n)$, Stirling number of the second kind } \\ \hline \end{array} \)
Source: https://www.careerbless.com/aptitude/qa/permutations_combinations_imp7.php#p1
\(S(k,n)\):
Source: https://www.wolframalpha.com/input/?i=stirling+number+second+kind
\(\begin{array}{|rcll|} \hline \text{Distribution of $5$ distinct Balls into $3$ identical Boxes} &=&\sum \limits_{i=1}^{3}S(5,i) \\ &=& S(5,1)+ S(5,2)+S(5,3) \\ &=& 1 + 15 + 25 \\ &=& 41 \\ \hline \end{array} \)
