Answer to Question #199329 in Statistics and Probability for Bahawal tahir

Question

(a) How many different positive three-digit whole numbers can be formed from the four digits 2, 6, 7, and 9 if any digit can be repeated?

(b) How many different positive whole numbers less than 1000 can be formed from 2, 6, 7, 9 if any digit can be repeated? 

(c) How many numbers in part (b) are less than 680 (i.e. up to 679)? 

(d) What is the probability that a positive whole number less than 1000, chosen at random from 2, 6, 7, 9 and allowing any digit to be repeated, will be less than 680?

Answer 1

a) Repetitions are allowed

$n=4, r=3$

$$n^r=4^3=64$$

b) Repetitions are allowed

$n=4, r=1$

$$n^r=4^1=4$$

$n=4, r=2$

$$n^r=4^2=16$$

$n=4, r=3$

$$n^r=4^3=64$$

Then

$$4+16+64=84$$

c) Numbers from $692$ to $699$

$$692, 696, 697, 699$$

Numbers from $722$ to $799$

$n=4, r=2$

$$n^r=4^2=16$$

Numbers from $922$ to $999$

$n=4, r=2$

$$n^r=4^2=16$$

Then numbers less than 680

$$84-(4+16+16)=58$$

d)

$$P(<680)=\dfrac{58}{84}=\dfrac{29}{42}\approx0.690476$$