Question

In a multi-state lottery, the player must guess which five of forty nine white balls numbered from 1 to 49 will be drawn. The order in which the balls are drawn does not matter. The player must also guess which one of forty-two red balls numbered from 1 to 42 will be drawn. How many ways can the player fill out a lottery ticket?

Short answer

The number of ways the player can fill out a lottery ticket is80089128.

Step-by-step solution

Step 1. Given Information.

Given in a multi-state lottery, the player must guess which five of forty nine white balls numbered from 1 to 49 will be drawn. The order in which the balls are drawn does not matter. The player must also guess which one of forty-two red balls numbered from 1 to 42 will be drawn. How many ways can the player fill out a lottery ticket. The number of ways the player can fill out a lottery ticket is to be determined.

Step 2. Calculation.

A permutation is when n objects are available and r are to be picked and arranged in a certain order and the number of permutations is given by $P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}$

A combination is when n objects are available and r are to be picked without arrangement and the number of combinations is given by $C\left(n,r\right)=\frac{n!}{\left(n-r\right)!r!}$

Five of the forty nine white balls can be selected in $C\left(49,5\right)$ ways.

One of the forty-two red balls can be selected in $C\left(42,1\right)$ ways.

Hence the total number of ways is given by:

$\begin{array}{l}C\left(49,5\right)\cdot C\left(42,1\right)=\frac{49!}{\left(49-5\right)!5!}\cdot \frac{42!}{\left(42-1\right)!1!}\\ C\left(49,5\right)\cdot C\left(42,1\right)=\frac{49!}{44!5!}\cdot \frac{42!}{41!1!}\\ C\left(49,5\right)\cdot C\left(42,1\right)=\frac{49\cdot 48\cdot 47\cdot 46\cdot 45\cdot 44!}{44!\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}\cdot \frac{42\cdot 41!}{41!\cdot 1}\\ C\left(49,5\right)\cdot C\left(42,1\right)=\frac{49\cdot 48\cdot 47\cdot 46\cdot 45}{5\cdot 4\cdot 3\cdot 2\cdot 1}\cdot \frac{42}{1}\\ C\left(49,5\right)\cdot C\left(42,1\right)=\frac{\text{9,61,06,95,360}}{120}\\ C\left(49,5\right)\cdot C\left(42,1\right)=80089128\end{array}$

Step 3. Conclusion.

Hence, the number of ways the player can fill out a lottery ticket is 80089128.