Question

How many ways are there to distribute 15 distinguishable objects into five distinguishable boxes so that the boxes have one, two, three, four, and five objects in them, respectively.

Short answer

There are 37837800 ways to distribute 15 distinguishable objects into five distinguishable boxes

Step-by-step solution

Step 1: Use the formula for integer

Formula for integer

\(\begin{array}{l}\frac{{n!}}{{{n_1}!{n_2}!..{n_k}!}} = \frac{{n!}}{{r!}}\\r! = {n_1}!{n_2}!..{n_k}!\end{array}\)

n is number of objects

r is number of objects in the box

Step 2: Solution of number ways 

Let’s, applied n and r values of inequality in integer formula

Distributing n distinguishable objects into r distinguishable boxes

Here,

\(\begin{array}{l}C\left( {n,r} \right) = \frac{{n!}}{{r!}}\\n = 15\\r = 1!,2!,3!,4!,5!\\C(15,(1!,2!,3!,4!,5!)) = \frac{{15!}}{{1!.2!.3!.4!.5!}}\\ = 37837800ways\end{array}\)