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?

7. Answer question 7 again for the case where the digits 2, 6, 7, 9 can not be repeated

Answer 1

Solution

a) Repetitions are allowed

$n=4, ~r=3$

$=n^r =4^3$

$=64$

(b) Repetitions are allowed

$n=4, ~r=1$

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

$n=4, ~r=2$

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

$n=4,~r=3$

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

Total number that can be formed

$=4+16+64$

$=84$

(c) Repetitions allowed.

1.Numbers from 692 to 699

$692,696,697,699$

2. Numbers from 722 to 799 $n=4, r=2$

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

3. Numbers from 922 to 999

$n=4,~ r=2$

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

Less than 680

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

(d) Probability that a number is less than 680

$P(X<680)=\dfrac {58}{84}\approx 0.69$

7. Repetition not allowed

(a) Possible samples

$n=5,r=3$

$=C_4^3= \binom{4}{3}=4$ samples

(b) Possible samples less than 1000

since $r=3$ all $4$ samples will be less than $1000$

(c) Samples less than 680

The samples will be as follows

$(2,6,7)~(2,6,9)~(2,7,9)~(6,7,9)$

All $4$ samples will be less than 680.

(d) Probability that a number is less than 680

$P(X<680)=\dfrac44=1$