Question

A security alarm requires a four-digit code. The code can use the digits 0–9 and the digits cannot be repeated. Which expression can be used to determine the probability of the alarm code beginning with a number greater than 7?

Answer 1

Answer:

$\text{required probability}=\frac{2\cdot 9\cdot 8\cdot 7}{\frac{10\cdot9\cdot 8\cdot 7\cdot 6!}{6!}}$

Step-by-step explanation:

We have total of 10 digits from 0 to 9

We need a four digit code and beginning with a number greater than 7

We have 2 choices for number greater than 7

For first digit we have 2 choices

Now, we are left with 9 choices because one is filled and digits should not repeat.

For second digit we have 9 choices

For third digit we will have 8 choices

And for fourth digit we will have 7 choices

Total number of choices are $^{10}P_4$

using $^{n}P_r=\frac{n!}{(n-r)!}$

$\text{required probability}=\frac{\text{favourable choices}}{\text{total choices}}$

$\text{required probability}=\frac{2\cdot 9\cdot 8\cdot 7}{\frac{10\cdot9\cdot 8\cdot 7\cdot 6!}{6!}}$

Cancel common terms from numerator and denominator we get:

$\text{required probability}=\frac{1}{5}$.