IIT JEE , AIEEE Forum - Mathematics , Algebra

debayudh_das

a seven digit number is formed wieh 1,2,3,4,5,6,7,8 and 9. find the probability that the number is divisable by 9

taruntanuj007

See the problemis a bit lenthy

Wat u do is first we coose 7 digits from 10 given and if ther sum is equal to a multiple of 9 then the number formed by those numerals regardless of their order will also be a multiple of 9. Try this if it doesnt work out i'll post the solutions

rik_mad

look, trying to find the sum to be multiple of 9 for 9c7 numbers (which is 36) is absolutely impractical.
u cud try this,
the smallest possible sum of seven numbers will be
1 + 2 + 3 + 4 + 5 + 6 + 7 = 28
and the largest possible sum is
3 + 4 + 5 + 6 + 7 + 8 + 9 = 42.
there is only one number between 28 and 42 which is divisible by 9 which is 36
36 hmmmm..which means 4 times 9
wat r the combinations of numbers which can give u nine
clearly they are
1 + 8
2 + 7
3+ 6
4 + 5
and 9 itself
in all there are 5 such elements possible
now u can't choose the first four elements as then ur total number of elements will be 8
so only 4c3(the frst four)= 4 combinations r possible which along with digit 9 will give 36.
now ur probabilty will become
7! * (4) / (9c7 * 7!)
where i hav used 7! for finding different permutations within the selected numbers
so probabilty is 4/36 = 1/9
simple !!

Asmita

But i got a different ans.

Total possible outcomes, n(S)=C(9,7)*7!=36*7!

Now 1+2+.........+9=9*10/2 i.e. a multiple of 9.

we've 2 take only 7 digits out of 9 i.e. we've 2 reject 2 digits.

For fav. outcomes those 2 rejected digits must give 9 when added. there r 4 such pair i.e. 1,8 /2,7 /3,6 /4,5

So fav. outcomes, n(E)=C(4,2)*7!=6*7!

So probability,P(E)=n(E)/n(S)=1/6

Plz tell me whether i m correct or not ?

rik_mad

asmita ur method is absolutely rite !! infact i find it better than mine !!

but instead of taking 4c2 combinations u must take 4c1 as u will be singling out only one pair of numbers ..not two !!

salute for a better method !

Asmita

Thanx 4 pointing out my mistake.

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