Question

20.* Four digit numbers are to be formed by using the digits
0,1,2,3,4,5. What is the number of such numbers if :
(a) Repetition is not allowed,
(b) Repetition is allowed,
(c) At least one digit is repeated ?​

Answer 1

a) The number of such numbers will be 5×5×4×3=300

Reason:-There might be confusion that why first it is not (6×5×4×3)...but this is only in the case if the digits do not contain 0.

As 0 is not possible at thousand's place because if it will at thousand's place then the number will become three digit .For example:-0345=345

,so there is only possibility of 5 numbers at the thousand's place

b)The total numbers in this case will be 5×6×6×6=1080

5 is taken as factor because at thousand's place ,we can't put 0(shown in above example,)

c) For numbers in which at least one-digit repeated=All digits repeated-no digit repeated

=>1080-300=780

Answer 2

Given :

Four digit numbers are to be formed by using the digits

0,1,2,3,4,5. What is the number of such numbers if :

(a) Repetition is not allowed

(b) Repetition is allowed

(c) At least one digit is repeated ?

To find :

• Repetition is not allowed
• Repetition is allowed
• At least one digit is repeated ?

Solution :

$\pink{(a)\:Repetition\:is\:not\:allowed}$

= $\sf{^6p_4 - ^5p_3}$ = ( 6 × 5 × 4 × 3 ) - ( 5 × 4 × 3 )

= 60 × ( 6 - 1 )

= $\blue{300}$

$\pink{(b)\:Repetition\:is\:allowed}$

= 1 st place in 5 ways ( no zero ), 2 nd place in 6 , 3 rd in 6 and 4 th also in 6.

Therefore, $\blue{5 × 6³}$

$\pink{(c)\:At\:least\:one\:digit\:is\:repeated\:?}$

= Total with repetiition - No Repetition

= 5 × 6³ - 300

= 1080 - 300

= $\blue{780}$