[Math] In how many ways can $10$ different books be distributed to $3$ students

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In how many ways can $10$ different books be distributed to $3$ students so that each student receives at least one book ?

Although I did solve by using Binomial Distribution, but just for curiosity –>

Is Sterling's Formula applicable for such problems ?

How can I apply Sterling's formula of 2nd kind here ?

My Question is also a possible duplicate of below question and I am very well aware of this method.

Distribution of 10 different books to three students.

I am just interested in knowing whether Sterling formula of 2nd kind applicable ?

Best Answer

Use inclusion/exclusion principle:

  • Include the number of ways to distribute $10$ books to at most $\color\red3$ students: $\binom{3}{\color\red3}\cdot\color\red3^{10}$
  • Exclude the number of ways to distribute $10$ books to at most $\color\red2$ students: $\binom{3}{\color\red2}\cdot\color\red2^{10}$
  • Include the number of ways to distribute $10$ books to at most $\color\red1$ students: $\binom{3}{\color\red1}\cdot\color\red1^{10}$

Hence the total number of ways is:

$$\sum\limits_{n=0}^{2}(-1)^{n}\cdot\binom{3}{3-n}\cdot(3-n)^{10}$$

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