How many ways are there to place 8 indistinguishable balls into four…
2019
How many ways are there to place 8 indistinguishable balls into four distinguishable bins?
- A.
\(70\) - B.
\(165\) - C.
\(8C_4\) - D.
\(8P_4\)
Attempted by 95 students.
Show answer & explanation
Correct answer: B
Key idea: Use the stars and bars method to count distributions of indistinguishable balls into distinguishable bins.
Represent the 8 indistinguishable balls as 8 stars.
Place 3 dividers (bars) to separate the stars into 4 bins. Together there are 8 stars + 3 bars = 11 symbols.
Choose positions for the 3 dividers among the 11 positions, which gives C(11,3) possible arrangements.
Compute C(11,3) = 11*10*9 / (3*2*1) = 165.
Therefore, the number of ways to place 8 indistinguishable balls into four distinguishable bins is C(11,3)=165.
Common pitfalls: C(8,4)=70 counts combinations of 4 chosen from 8 and is not the distribution count; 8P4 counts ordered selections and is not applicable because the balls are indistinguishable.