Question

1. in the california state lottery, you must select six numbers from fifty - two numbers to win the big prize. the numbers do not have to be in a particular order. what is the probability that you will win the big prize if you buy one ticket?

$_{n}c_{r}=\frac{n!}{n!r!}$
$=\frac{52!}{5!6!}$

Explanation:

Step1: Identify the combination formula

The problem involves selecting 6 numbers from 52, where order doesn't matter. The combination formula is \( _nC_r=\frac{n!}{r!(n - r)!} \), where \( n = 52 \) and \( r = 6 \).

Step2: Calculate the number of possible combinations

Substitute \( n = 52 \) and \( r = 6 \) into the formula:

$$ _{52}C_6=\frac{52!}{6!(52 - 6)!}=\frac{52!}{6!×46!} $$

Simplify \( 52! = 52×51×50×49×48×47×46! \), so:

$$ \frac{52×51×50×49×48×47×46!}{6!×46!}=\frac{52×51×50×49×48×47}{6×5×4×3×2×1} $$

Calculate the numerator: \( 52×51 = 2652 \), \( 2652×50 = 132600 \), \( 132600×49 = 6497400 \), \( 6497400×48 = 311875200 \), \( 311875200×47 = 14658134400 \)
Calculate the denominator: \( 6×5 = 30 \), \( 30×4 = 120 \), \( 120×3 = 360 \), \( 360×2 = 720 \), \( 720×1 = 720 \)
Now divide: \( \frac{14658134400}{720}=20358520 \)

Step3: Determine the probability

There is 1 winning combination. So the probability \( P=\frac{1}{_{52}C_6}=\frac{1}{20358520} \)

Answer:

The probability is \(\frac{1}{20358520}\)