If your solution to stated problem does not match the given solution, you should self-critique per instructions at

http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm.

Your solution, attempt at solution. 

If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it.  This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

002.  Permutations, combinations, rearranging letters of words.

Question:  `q001.  Note that there are 9 questions in this assignment.

The situation is the same as in q_a_#1:

Imagine three boxes:

• The first contains a set of billiard balls numbered 1 through 15.
• The second contains a set of letter tiles with one tile for each letter of the alphabet.
• The third box contains colored rings, one for each color of the rainbow (these colors are red, orange, yellow, green, blue, indigo and violet, abbreviated ROY G BIV).

If we choose three letter tiles from the second box and lay them in a row, in the order chosen, then how many three-letter 'words' are possible? 

(Note that we count any string of 3 letters as a word, whether it appears in the dictionary or not)

Your solution: 

Your solution:

Confidence Rating:

Given Solution:
 

There are 26 choices for the first tile chosen, 25 for the second and 24 for the third.  The number of possible three-letter words with 3 distinct letters of the alphabet is therefore 26 * 25 * 24.

Self-critique (if necessary):

Self-critique rating:

Question: 

  `q002.  If we choose three letter tiles from the second box, then how many unordered collections of three letters are possible?

Your solution: 

Your solution:

Confidence Rating:

Given Solution:
 

  If the 3-tile collections are unordered there are only 1/6 as many possibilities as for the ordered collection, since there are 3 * 2 * 1 = 6 orders in which any given collection might have been chosen.

Since there are 26 * 25 * 24 ways to choose the 3 tiles in order, there are thus 26 * 25 * 24 / 6 possibilities for unordered choices.

STUDENT QUESTION

I think I could still use a bit of explanation as to why you do the 3*2*1 for an unordered
collection. I understand about making the three choices but the 6 orders in choosing the given collection confuses me.

INSTRUCTOR COMMENT

For example you could choose the c, m and q tiles in any of the six orders cmq, cqm, mcq, mqc, qcm, qmc. 

• If you are making ordered choices, all six would be different.

• But if choices are unordered, they all contain the same letters so these six ordered choices would all be the same.

Clearly the same is true for any collection of 3 letters.

So there are 6 times more ordered choices than unordered choices.

Self-critique (if necessary):

Self-critique rating:

Question: 

  `q003.  If we choose two balls from from the first box of the original problem, and do so without replacing the first ball chosen, we can get totals like 3 + 7 = 10, or 2 + 14 = 16, etc.. 

• How many of the possible unordered outcomes give us a total of less than 29?

STUDENT QUESTION

I guess what I am unsure of when it comes to dividing these out by the possible choices is the
way you arrive at 2 for this problem and 6 for the last one. I realize that you are making two choices each time, but for the last problem, you are only making three, then two then one and they are unordered, but for some reason it’s not clicking in my brain.

INSTRUCTOR RESPONSE

If you choose balls 3 and 7, then they could be chosen in the order 3,7 or in the order 7,3.

The unordered collection is the same, no matter which ball is chosen first. 

So a single unordered collection of two objects corresponds to 2 different ordered choices.

Your solution: 

Confidence Rating:

Given Solution: 

The smallest possible total would be 1 + 2 = 3 and the greatest possible total would be 14 + 15 = 29.  Thus the only way to get a total of 29 is to have chosen 14 and 15, in either order (i.e., either 14 first and 15 second, or 15 first and 14 second).

• Thus out of the 15 * 14 / 2 = 105 possible unordered combinations of two balls, only one gives us a total of at least 29. 
• The remaining 104 possible combinations give us a total of less than 29.

This problem illustrates how it is sometimes easier to analyze what doesn't happen than to analyze what does.  In this case we were looking for totals less than 29, but it was easier to find the number of totals that were not less than 29.  Having found that number we easily found the number we were seeking.

Self-critique (if necessary):

Self-critique Rating:

Question: 

  `q004.  If we place each object in all the three boxes (one containing 15 numbered balls, another 26 letter tiles, the third seven colored rings) in a small bag and add packing so that each bag looks and feels the same as every other, and if we then thoroughly mix the contents of the three boxes into a single large box before we pick out two bags at random:

• How many of the possible combinations will include two rings?
• How many of the possible combinations will include two tiles?
• How many of the possible combinations will include a tile and a ring?
• How many of the possible combinations will include at least one tile?

Your solution: 

Your solution:

Confidence Rating:

Given Solution:
 

There are a total of 7 rings. 

• There are thus 7 ways the first bag could have contained a ring, leaving 6 ways in which the second bag could have contained a ring. 
• It follows that 7 * 6 / 2 = 21 of the possible combinations will contain 2 rings (note that we divide by 2 because each combination could occur in two different orders).

Reasoning similarly we see that there are 26 ways the first bag could have contained a tile and 25 ways the second bag could have contained a tile, so that there are 26 * 25 / 2 = 325 possible combinations which contain 2 tiles.

Since there are 26 tiles and 7 rings, there are 26 * 7 / 2 = 91 possible two-bag combinations containing a tile and a ring.

There are a total of 15 + 26 + 7 = 48 bags, so the total number of possible two-bag combinations is 48 * 47 / 2. 

• Since 15 + 7 = 22 of the bags do not contain tiles, there are 22 * 21 / 2 two-bag combinations with no tiles. 
• The number of possible combinations which do include at least one tile is therefore the difference 48 * 47 / 2 - 22 * 21 / 2 between the number of no-tile combinations and the total number of possible combinations.

Self-critique (if necessary):

Self-critique rating:

Question: 

  `q005.  Suppose we have mixed the contents of the three boxes as described in the preceding problem. 

If we pick five bags at random, then in how many ways can we get a ball, then two tiles in order, then a ring, then another ball, in that order?

Your solution: 

Your solution:

Confidence Rating:

Given Solution:
 

There are five objects to choose.  We apply the fundamental counting principle to each of the five choices:

• There are 15 bags containing balls, so there are 15 ways to get a ball on the first selection.
• If a ball is chosen on the first selection, there are still 26 bags containing tiles when the second selection is made.  So there are 26 ways to get a tile on the second selection.
• At this point there are 25 tiles so there are 25 ways to get a tile on the third selection.
• There are still 7 rings from which to select, so that there are 7 ways the fourth choice can be a ring.
• Since 1 ball has been chosen already, there are 14 ways that the fifth choice can be a ball.

To get the specified choices in the indicated order, then, there are 15 * 26 * 25 * 7 * 14 ways.

STUDENT COMMENT

This was much easier to grasp than the previous problem for some reason.

INSTRUCTOR RESPONSE

That's because you're thinking about the problems and asking good questions. This is the key to success.

Keep up the good work.

Self-critique (if necessary):

Self-critique rating:

Question: 

  `q006.  Suppose we have mixed the contents of the three boxes as in the preceding problems. 

If we pick five bags at random, then in how many ways can we get two balls, two tiles and a ring in any order?

Your solution: 

Your solution:

Confidence Rating:

Given Solution:
 

This time the order in which the choices are made doesn’t matter. 

• There are 15 * 14 possible outcomes when 2 balls are chosen in order, and 15 * 14 / 2 possible outcomes when the order doesn't matter.
• There are similarly 26 * 25 / 2 possible outcomes when 2 tiles are choseb without regard for order.
• There are 7 possible choices for the one ring.

Thus we have [ 15 * 14 / 2 ] * [ 26 * 25 / 2 ] * 7 ways in which to make an unordered choice of 2 balls, 2 tiles and a ring.

Another way to get the same result is to start with the 15 * 26 * 25 * 7 * 14 ways to choose the 2 balls, 2 tiles and one ring in a specified order, as shown in the last problem. 

• Whichever 2 tiles are chosen, they could have been chosen in the opposite order, so if the order of tiles doesn't matter there are only half as many possible outcomes--i.e., 15 * 26 * 25 * 7 * 14 / 2 possibilities if the order of the tiles doesn't matter that the order of the balls does. 

• If the order of the balls doesn't matter either, then we have half this many, or 15 * 26 * 25 * 7 * 14 / ( 2 * 2) ways.  It should be easy to see why this expression is identical to the expression [ 15 * 14 / 2 ] * [ 26 * 25 / 2 ] * 7 obtained by the first analysis of this problem.

Self-critique (if necessary):

Self-critique rating:

Question: 

  `q007.  Suppose we have mixed the contents of the three boxes as described above. 

• If we pick five bags at random, then in how many ways can we get a collection of objects that does not contain a tile?

Your solution: 

Your solution:

Confidence Rating:

Given Solution:
 

Of the 48 bags, 26 do and 22 do not contain a tile. 

• If we pick five bags at random, then there are 22 * 21 * 20 * 19 * 18 ordered ways in which the five bags could all contain something besides a tile.
• Any given collection of five bags could have been chosen in any of 5 * 4 * 3 * 2 * 1 orders.
• There are therefore 22 * 21 * 20 * 19 * 18 / (5 * 4 * 3 * 2 * 1) possible unordered collections of five bags.

GOOD STUDENT COMMENT:

Ok, so in the previous questions, when deciding what number to divide by, you always multiply the number of orders that
a bag could have been chosen. So when dividing by 2, it was because you were only choosing two bags (2*1=2). But in
this case, there are 5 bags to choose from (5*4*3*2*1). Then you divide the number of ordered ways by this number.

INSTRUCTOR RESPONSE: 

Very good.  You stated that well.
 

Self-critique:

Self-critique rating:

Question: 

  `q008.  Suppose the balls, tiles and rings are back in their original boxes.  If we choose three balls, each time replacing the ball and thoroughly mixing the contents of the box, then two tiles, again replacing and mixing after each choice, how many 5-character 'words' consisting of 3 numbers followed by 2 letters could be formed from the results? 

Your solution: 

Your solution:

Confidence Rating:

Given Solution:
 

Since the order of the characters makes a difference when forming 'words', the order of the choices does matter in this case. 

• We have 15 balls from which to choose, so that if we choose with replacement there are 15 possible outcomes for every choice of a ball.

• Similarly there are 26 possible outcomes for every choice of a tile.

• Since we first choose 3 balls then 2 tiles, there are 15 * 15 * 15 * 26 * 26 possible 5-character 'words'.

Question: `q009.  Suppose we choose three balls from one through fifteen, without replacing the balls we choose, and place the three chosen balls in order from left to right.  We write down the numbers on these balls in order, forming a larger number which can be 3, 4, 5 or 6 digits long (for example if the numbers are 1, 7 and 12 the resulting number would be 1712).

Now, in order to get a 3-digit number, the each of the three balls chosen must have a 1-digit number.  Thus each of the first three balls must show one of the digits 1 through 9.

Some of the following questions are more challenging than others.  Give your best thinking on each one:

Answer the following in a similar manner:

• What has to happen in order to end up with a six-digit number?

• What has to happen in order to end up with a five-digit number?

• What has to happen in order to end up with a number having two 3's?

• Are there any numbers that could be obtained in more than one way (i.e., by selecting two or more different sets of three balls)?

Answer the following:

• How many different three-digit numbers are possible?

• How many different six-digit numbers are possible?

• How many possible numbers contain no 1's?

• How many possible numbers contain four 1's?

Your solution:

Confidence Rating:

Self-critique Rating: