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Math circle level problem on binomial coefficients
Problem 1In how many ways could five different envelopes be distributed into three mailboxes?
Solution 1
I will consider consequently the cases when where is no envelopes in the box #3;
when there is 1 envelope in the box #3;
when there is 2 envelopes in the box #3;
when there is 3 envelopes in the box #3;
when there is 4 envelopes in the box #3;
when there is 5 envelopes in the box #3.
For each of these cases, I will calculate the number of different distributions of the envelopes in three mailboxes, as follows.
a) 0 envelopes in the box #3: = + + + + + = 1 + 5 + 10 + 10 + 5 + 1 = 32 ways.
b) 1 envelope in the box #3: = 5*( + + + + ) = 5*(1 + 4 + 6 + 4 + 1) = 80 ways.
c) 2 envelopes in the box #3: = 10*( + + + ) = 10*(1 + 3 + 3 + 1) = 80.
d) 3 envelopes in the box #3: = 10*( + + ) = 10*(1 + 2 + 1) = 40 ways.
e) 4 envelopes in the box #3: = 5*( + ) = 5*2 = 10 ways.
f) 5 envelopes in the box #3: = 1 way.
The total is the sum 32 + 80 + 80 + 40 + 10 + 1 = 243 ways.
Answer. 243 ways.
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The formulas in this post are SELF-EXPLANATORY.
If you have questions or if you need explanations, look into the formulas until they tell you the whole story.
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Notice that 243 = , and it is not eventually.
There is another way to calculate it, which gives the same result, but is much shorter and much more elegant.
Solution 2
You need to consider the binomial expansion of as the sum
= sum of all with the coefficients , i + j + k = 5.
Each particular term with i + j + k = 5 "marks" some possible particular distribution
of envelopes in the three boxes called "x", "y" and "z",
and the coefficient at this term represents the "multiplicity" of the given concrete distribution
(i,j,k) in the boxes x, y, and z.
The number of all possible distributions is the sum of all coefficients , and it is equal to
the value of at x= 1, y= 1, z= 1, which is exactly = = 243.
This problem is for an advanced Math circle / Math Olympiad level.
Therefore, I will not go further into details - the idea is just presented very clearly for an adequate person.
My lessons on Binomial Theorem, Binomial Formula, Binomial Coefficients and Binomial Expansion in this site are
- Binomial Theorem, Binomial Formula, Binomial Coefficients and Binomial Expansion
- Remarkable identities for Binomial Coefficients
- The Pascal's triangle
- Solved problems on binomial coefficients
- Solving equations that include binomial coefficients and numbers of permutations
- Math circle level problem on binomial coefficients (this lesson)
- OVERVIEW of lessons on Binomial Expansion, Binomial coefficients and the Pascal's triangle
Use this file/link ALGEBRA-II - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-II.
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