Q.1.31 - A First Course in Probability

Question

If $8$identical blackboards are to be divided among $4$schools, how many divisions are possible? How many of each school must receive at least $1$a blackboard?

Step-by-Step Solution

Answer

Interpret the question as to the number of positive and nonnegative results of an equation.

If every school gets a blackboard =$5$possibilities

If not every school needs to get a blackboard =$105$possible distributors.

Step 1 Given Information.

Let 8 identical blackboards are to be divided among 4 schools. 

Step 2 Explanation.

Compute the number of non-negative integer solutions of the following equation:

$x_1+x_2+x_3+x_4=9$.

where$x_1$represents the number of blackboards in the school$1,x_2$ represents the number of blackboards in school $2$...etc.

If every school has to get at least one blackboard, the number of nonpositive integer solutions to the following negation:

$x_1+x_2+x_3+x_4=8$

$\left(\begin{array}{l}7\\ 3\end{array}\right)=35$

For positive solutions,

$y_1+y_2+y_3+y_4=8+4=12$

And that is, according to previous consideration:

$\left(\begin{array}{c}11\\ 3\end{array}\right)=165$