Assignment 3 part I 6 polzita 1. A fast food zcs‹aurant is ofEedng any fotz differed items fzoct th•ir menu for $6.?i0. They claim that liters arc 330 dififétent choices, How miuty ilmns am in IM...

Question 1: A fast food restaurant is offering any four different items from their menu for
$6.50. They claim that there are 330 different choices. How many items are in the menu?
Solution: We can choose 4 items from a list of items. It is given that the total number of ways to
choose 4 items is 330. Thus,
3304 C
n
330
!4)!4(
!

n
n

Solving this we will get, n = 11.
Question 2: How many different positive integers n can be named using the digits 3, 3, 4, 5,
5, 6, 7 if we want n to exceed 5,000,000?
Solution: We must use all of the numbers in order to get a 7 digit number. We also must have one of
the 5’s, the 6 or the 7 first in order to exceed 5,000,000. Therefore there are 4 choices for the first
digit, 6 choices for the second digit, 5 choices for the third digit, 4 choices for the fourth digit, etc.
Therefore there are 4 · 6 · 5 · 4 · 3 · 2 · 1 = 2880 ways to arrange these digits. However, there are
two 3’s and two 5’s so we need to divide by 2! · 2!
Therefore, 720
4
2880
!2!.2
1.2.3.4.5.6.4
 numbers possible
Question 3: For an examination, a professor has chosen 4 questions from chapter 1, 5
questions from chapter 2, and 3 questions from chapter 3.
a) How many different lists of questions can she make from this set of questions?
Solution: Considering the fact that there would be one question per chapter in the
examination, the total number of questions are 12 in number and we will make a paper with
three questions.
These 12 questions can be arranged in following way:
12 x 11 x 10 = 1320 ways
b) How many different lists of questions can she make if the questions from chapter 1,
must be presented first, followed by the questions from chapter 2, and ending with
the questions from chapter 3?
Solution: Now it is said that the order is fixed, chapter 1 at beginning followed by 2 and 3.
So, the number of questions possible at first position are only 4, for second place its 5 and
for third place its 3.
Thus, it would be written as:
11
1
3
1
5
1
4
3
12

  CCC
C

Question 4: How many ways are there to place 12 marbles of the same size in five different
containers if
a) All the marbles are black?
Solution: The number of ways to place this = 12 x 11 x 10 x 9 x 8 =
b) All the marbles are black and no container is empty?
c) Each marble is of different color?
Solution: The number of ways to place are
12
C5 = 792 ways
Question 5: Explain why, if we select 10 points within an equilateral triangle with side 1,
there must be at least two whose distance apart is less than or equal to 1/3.
Solution: Let us divide equilateral triangle ABC to smaller equilateral triangles as shown in
figure, in which each small equilateral triangle's side length is 1/3. So any two points in same
small triangle are apart from the other at most 1/3. If 10 points are chose from triangle ABC, then
at least two of points fall into same small triangle, so their distance apart is at most...