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MATA67H3 (3)

Richard Pancer (2)

Lecture 4

Department

MathematicsCourse Code

MATA67H3Professor

Richard PancerLecture

4This

**preview**shows pages 1-3. to view the full**14 pages of the document.**History of Pascal’s Triangle

Pascal was not the ﬁrst to discover the triangle of binomial coef-

ﬁcients but was given credit because of how he related it to his

work with probability and expectation.

The triangle may have ﬁrst appeared more than 300 years earlier

during the 11th century in China:

1

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Distributing Objects

Example. How many ways can 20 different diplomats be as-

signed to 5 different continents?

Solution. Rephrase the problem as an arrangement we already

know.

Q. What if each continent needs to have 4 diplomats each?

A.

Example. How many ways are there to distribute 20 identical

chocolate bars and 15 identical sticks of gum to 5 children?

Solution.

2

This really is the password problem

5^20

5^20

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Example. How many integer solutions are there to the equation

x1+x2+x3+x4= 12 with xi0?

Solution.

Q. What if we require that each xi1?

A.

Theorem. The number of ways to distribute ridentical objects

into ndistinct boxes with at least one object in each box is C(n

1,r1).

Proof. We need to place nof the robjects amongst the nboxes

leaving us with rnobjects to distribute into the nboxes.

3

(0 included)

C(15, 3) = C(15, 12)

C(11, 3)

c(r-1, n-1)

C(r-n+(n-1), (n-1))

Another way

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