201k views
2 votes
How many ways can you divide nine children into a team of 4, a team of 3, and a team

of 2?

Ana writes down all the numbers 1, 2, . . . , n, for some positive integer n. In doing so,
she writes a total of 600 digits. Find the positive integer n.

User Azpiri
by
7.7k points

1 Answer

6 votes

Answer:

If we want to divide a group of N objects into K objects; we have


c = (N!)/((N - K)!*K!)

combinations

For the first group, N = 9 and K = 4


c1 = (9!)/(5!*4!) = (9*8*7*6)/(4*3*2) = 126

For the second group we have N = 5 (cause we already took 4 of them) and K = 3


c2 = (5!)/(2!*3!) = 10

For the last group we have N =2 and K = 2, so there is only one possible combination, c3 = 1 then we have that the total number of combinations is:

C = c1*c2*c3 = 126*10*1 = 1260 possible divisions.

b)

numbers with one digit = 9, in those 9 numbers we have 9 digits.

numbers with two digits; (99 - 10 = 89) numbers, in those 89 numbers we have 2*89 = 178 digits, plus the 9 of before we have 178 + 9 = 183.

Now, for each number of 3 digits we have 3 digis obviusly, so we can write:

N*3 = 600 - 183

and N is the number of 3 digits number needed.

N = 317/3 = 139

So the 139-th number of 3 digits, this is:

100 + 139 - 1

The minus one goes because 100 is already a 3 digit number:

100 + 139 - 1 = 238

238 is the positive integer n.

User TheSean
by
8.1k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories