# ISEE Middle Level Math : How to find the missing part of a list

## Example Questions

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### Example Question #1 : Sets

What are the next two numbers of this sequence?

Explanation:

The sequence is formed by alternately addingÂ  and addingÂ  to each term to get the next term.

Â andÂ  are the next two numbers.

### Example Question #1 : How To Find The Missing Part Of A List

Define two sets as follows:

Which of the following is a subset ofÂ Â ?

Each of the sets listed is a subset ofÂ .

Each of the sets listed is a subset ofÂ .

Explanation:

We demonstrate that all of the choices are subsets ofÂ .

Â is the intersection ofÂ Â andÂ Â - that is, the set of all elements ofÂ bothÂ sets. Therefore,Â

Â itself is one of the choices; it is a subset of itself. The empty setÂ Â is a subset ofÂ everyÂ set. The other two sets listedÂ comprise only elements fromÂ , making them subsets ofÂ .

### Example Question #2 : How To Find The Missing Part Of A List

Let the universal setÂ Â be the set of all positive integers. Also, define two sets as follows:

Which of the following is an element of the setÂ Â ?

Explanation:

We are looking for an element that is in theÂ intersection of Â andÂ Â - in other words, we are looking for an element that appearsÂ inÂ both sets.

Â is the set of all multiples of 8. We can eliminate two choices as not being inÂ Â by demonstrating that dividing eachÂ by 8 yields a remainder:

Â is the set of all perfect square integers. Â We can eliminate two additional choices as not being perfect squares by showing that each is between two consecutive perfect squares:

This eliminates 352 and 336. However,Â

.

ItÂ is also a multiple of 8:

Therefore,Â .

### Example Question #3 : How To Find The Missing Part Of A List

Define two sets as follows:

Which of the following is not an element of the setÂ Â ?

Explanation:

Â is theÂ unionÂ ofÂ Â andÂ , the set of all elements that appear inÂ either set. Therefore, we are looking to eliminate the elements inÂ Â and those inÂ Â to find the element in neither set.

Â is the set of all multiples of 8. We can eliminate two choices as mulitples of 8:

, soÂ

, soÂ

Â

Â is the set of all perfect square integers. We can eliminate two additional choices as perfect squares:

, soÂ

, soÂ

Â

All four of the above are therefore elements ofÂ .

Â

420, however is in neither set:

, soÂ

andÂ

, soÂ

Therefore, Â , making this the correct choice.

### Example Question #4 : How To Find The Missing Part Of A List

SevenÂ students are running for student council; each member of the student body will vote for three. Derreck does not want to vote for Anne, whom he does not like.Â How many ways can he cast a ballot so as not to include Anne among his choices?

Explanation:

Derreck is choosing three students from a field of six (seven minus Anne) without respect to order, making this a combination. He hasÂ Â ways to choose. This is:

Â

Derreck has 20 ways to fill the ballot.

### Example Question #5 : How To Find The Missing Part Of A List

Ten students are running for Senior Class President.Â Each member of the student body will choose four candidates, and mark them 1-4 in order of preference.Â

How many ways are there to fill out the ballot?

Explanation:

Four candidates are being selected from ten, with order being important; this means that we are looking for the number of permutations of four chosen from a set of ten. This is

There are 5,040 ways to complete the ballot.

### Example Question #1 : How To Find The Missing Part Of A List

The juniorÂ class elections have fourÂ students running for President, five running for Vice-President, fourÂ running for Secretary-Treasurer, and seven running for Student Council Representative. How many ways can a student fill out a ballot?

Explanation:

These are fourÂ independent events, so by the multiplication principle, the ballot can be filled outÂ Â ways.

### Example Question #7 : How To Find The Missing Part Of A List

The sophomore class elections have six students running for President, five running for Vice-President, and six running for Secretary-Treasurer. How many ways can a student fill out a ballot if he is allowed to select one name per office?

Explanation:

These are three independent events, so by the multiplication principle, the ballot can be filled outÂ Â ways.

### Example Question #8 : How To Find The Missing Part Of A List

Ten students are running for Senior Class President.Â Each member of the student body will choose fiveÂ candidates, and mark them 1-5 in order of preference.Â

Roy wants Mike to win. How many ways can Roy fill out the ballot so that Mike is his first choice?

Explanation:

Since Mike is already chosen, Roy is in essence choosing four candidates from nine, with order being important. This is a permutation of four elements out of nine. The number of these is

Roy can fill out the ballot 3,024 times and have Mike be his first choice.

### Example Question #1 : How To Find The Missing Part Of A List

Find the missing part of the list: