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How many eight digit numbers are there, that contain a 5 and a 6____________? please explain!

Ans: 8486912
asked in Combinatory by Veteran (11.5k points)   | 53 views

I suppose this should be the answer.

The negation of the statement:

Contains a 5 an a 6

is:

Thats includes following things:

1. Neither 5 nor 6

2. Contains only 5

3. Contains only 6

-------------------------------------------------------

1. Neither 5 nor 6 = 7 * 87 = 14680064

-------------------------------------------------------

2. Contains only 5(  but not 6 )

We use the methodology of contains atleast single 5.

So, Contains only 5 = 8 * 97 - 7 * 8= 23583688

-------------------------------------------------------

3. Contains only 6 (but not 5)

 = 23583688

-------------------------------------------------------

4. Now, total possible numbers without any restrictions = 9 * 107

 

Thus, answer = statement(4) - [statement(1) + statement(2) + statement(3)] = 28152560

1 Answer

+1 vote
number of numbers which contain a 5 and a 6
_ _ _ _ _ _ _ _
choose any two places from above 8 places ,arrange one 5 and one 6.  and each remaining 6 places have 10 possibilities (any of0...9 numbers can be present)
so 8C2 * 2!*10^6
but these arrangements also contain 0 in the first position. we need to subtract those numbers
0 _ _ _ _ _ _ _
7C2*2!*10^5 numbers contain 0 in the first position in 8C2 * 2!*10^6 arrangements
so subtract them
8C2*2!*10^6 - 7C2*2!*10^5 = 51800000 shud be the answer
answered by Boss (9.4k points)  

Can you solve it again by conidering that it's given as exactl one 5 and exactly one 6?

is it
8C2*2!*8^6  -  7C2*2!*8^5 

yes.
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