Question

1. (a) (i) How many different six-digit natural numbers may be formed

from the digits 2, 3, 4, 5, 7 and 9 if digits may not be repeated?

(ii) How many of the numbers so formed are even?

(iii) How many of the numbers formed are divisible by 3?

(iv) How many of the numbers formed are less than 700,000?

(b) JACK MURPHY’s seven character password consists of four let-

ters chosen from the ten letters in his name (all upper case) and

three digits; for example, JA01MU2 or P8YCK82.

(Note that the first example does not contain

repetition (of either

letters or digits)

and the second example has repetition.)

Out of these 10 different letters and the ten digits from 0 to 9,

how many arrangements of a password containing four letters and

three digits may be made in each of the following cases?

(i) There is no repetition (as in the first example above).

(ii) Repetition is allowed (as in the second example above).

(iii) There is no repetition and no two digits can be next to each

other; for example JA1C2K3 is allowed, but J1A23CK is not

because the 2 and 3 are beside each other.

(iv) The password consists of four letters followed by three digits

and there is no repetition.

(v) The password consists of four letters followed by three digits

and repetition is allowed.

Thanks

Answer 1

1. (a) (i) How many different six-digit natural numbers may be formed from the digits 2, 3, 4, 5, 7 and 9 if digits may not be repeated =6! =720

(ii) There are two 2 even digit

So, Number of even numbers formed =2 * 5! = 240

(iii) For a number to be divisible by 3, sum of its digits should be divisible by 3. Sum of digits= 2+3+4+5+7+9 =30.

So all numbers are divisible by 3.

(iv) Numbers of number formed is less than 700,000 = 4*5! = 480

b) (i) Password with no repetition =7! * (₁₀P₄) *(₁₀P₃)

= 5040 * ((10*9*8*7) + (10*9*8))

= 5040* 5760 = 29,030,400

(ii) Password with Repetition= 7! *( (10*10*10*10) +(10*10*10))

= 55,440,000

(iv) The password consists of four letters followed by three digits and there is no repetition.

= (₁₀P₄)+ (₁₀P₃) =10*9*8*7 + 10*9*8

= 5760

(v) The password consists of four letters followed by three digits and repetition is allowed.

= (10⁴) +(10³) = 11000