Question

number of ways to reorder the letters of the word \parallel\ =

number of ways to distribute 8 indistinguishable objects into 3 distinguishable boxes =

number of ways to distribute 6 distinguishable objects into 3 distinguishable boxes so that there are 2 objects in each box =

Explanation:

Step1: Count letters and duplicates

Word "PARALLEL" has 8 letters: 3 Ls, 2 As, 1 P, 1 R, 1 E.

Step2: Apply permutation formula

Formula for permutations of multiset: $\frac{n!}{n_1!n_2!...n_k!}$ where $n=8$, $n_1=3$ (L), $n_2=2$ (A)
$\frac{8!}{3!2!} = \frac{40320}{6 \times 2} = 3360$

Step3: Stars and bars formula

For indistinct objects, distinct boxes: $\binom{n+k-1}{k-1}$, $n=8$, $k=3$
$\binom{8+3-1}{3-1} = \binom{10}{2} = \frac{10!}{2!8!} = 45$

Step4: Distribute distinct objects

First choose 2 for box1: $\binom{6}{2}$, then 2 for box2: $\binom{4}{2}$, last 2 for box3: $\binom{2}{2}$
$\binom{6}{2} \times \binom{4}{2} \times \binom{2}{2} = 15 \times 6 \times 1 = 90$

Answer:

Number of ways to reorder the letters of the word "PARALLEL" = 3360
Number of ways to distribute 8 indistinguishable objects into 3 distinguishable boxes = 45
Number of ways to distribute 6 distinguishable objects into 3 distinguishable boxes so that there are 2 objects in each box = 90