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Question

Dear Mitch,

I have a question; it's not a criticism even though it may seem like one.

Recently you gave problems involving the dyeing of Easter eggs.  The questions included a rule: any two colors can be mixed together to form a new color but no more than two colors were allowed to be mixed with each other for this purpose.  So, any pair of original colors could be combined, but three or more in one mix could not.

You said that it's a very common question on standardized tests like the SAT exam, but I don't think I have seen it on any practice test even though I've been going through the review books. Can you tell me if I'm just not catching it?

 

Sincerely,

Jennifer B.

Naples, Italy

Answer

Dear Jennifer B.,

When I refer to a kind of question as a very popular one on standardized tests, I'm referring to either the underlying concept of the problem or a particular fact pattern that mirrors the fact patterns of other questions.  Sometimes, questions that may initially seem to be unrelated actually rely on the exact same math ideas.

Here, as I think we mentioned, we have a question that utilizes the idea of combinations of two separate entities (different items) that can be brought together to form one "Event" or "Union".

Mathematically 'famous', (i.e., commonly seen) examples include: Questions involving "handshakes", questions involving "kisses", questions involving "hugs", and questions involving sets of teams from a league of teams that pair up to play a single game at a time. Finally, sometimes on a standardized test one encounters a problem involving sets of wine glasses 'clinked' in celebration or, as we have here, combinations of paint colors. Still, the most common form is this: handshakes of people attending a meeting.

Example: Seven people attend a meeting, and they each shake hands with each other once. Question: How many handshakes will take place?

A) 7

B) 14

C) 21

D) Not enough information provided

Method: To represent the seven people attending the meeting, draw seven blanks across a sheet of paper, and number the blanks from 1 to 7, like this:

___  ___  ___  ___  ___  ___  ___

   1     2      3      4      5     6      7

One step at a time, we figure out how many unique handshakes each person is capable of making (by 'unique' we mean that if two people are so thrilled to see each other that they insist on shaking each other's hand every few minutes, they will still count as one unique combination). Keeping in mind the simple fact that when person A shakes person B's hand, it is the same event as person B shaking person A's hand, and so we must be careful to find the pattern that assures we do not repeat events simply because the order in which they appear is different. TWO PEOPLE SHAKE HANDS, AND IT IS A SIMULTANEOUS OCCURRENCE, REGARDLESS OF WHO APPROACHES FIRST.

QUESTION: How many people's hands can the first person shake?

ANSWER: 6

(Not 7, because that would require shaking his own hand)

So, you write a 6 up there in the first person's blank.

QUESTION: How many people's hands can the second person shake?

The other 6?

NO!,  because he (or she) has already had a handshake with one of them, person # 1.

The answer is 5.

QUESTION: How many people's hands can the fifth person shake?

ANSWER: 4 (Remember that they have to be 'new' hands, i.e., not hands of people who have shaken hands with person number four already.)

QUESTION: How many people's hands remain for the fourth person to shake?

ANSWER: 3

QUESTION: How many hands remain for the third person to shake?

ANSWER: 2

QUESTION: How many for the second person?

ANSWER: 1

How many for the first person?

ANSWER: 0, because everyone has already shaken his hand. (So he's been paired up with everyone else there already.)

So the pattern is this:

6    5   4   3      1   0

1    2    3   4    5   6   7

Then you simply add up the handshakes:

6 + 5 + 4 + 3 + 2 +1+ 0

(HINT: In a multi-step addition problem like this, it is often easiest to work from the small numbers to the larger ones, which here would mean do the addition from right to left, so that you spend more of your time working with small numbers than larger ones and will be less likely to make errors.)

SO: 0 + 1 + 2 + 3 + 4 + 5 + 6 =

                3 +      7 +      11 =

                     10     +      11 =

                            21

Our original question involved paint blends in combinations of two for the purpose of Easter Egg dyeing. It is the same math concept. So, seven different colors which are allowed to be mixed into combos of two will make 21 new combinations, just like the 21 handshakes (plus the original seven colors).

Hope this helps!

Happy Holiday,

Mitch