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course Mth 152
9/15 10
Question: `q001. Note that there are 8 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 third 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)
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
possible three-letter words with 3 distinct letters of the alphabet is therefore 26 * 25 * 24
3
Ok
`q002. If we choose three letter tiles from the third box, then how many unordered collections of three letters are possible?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
26 * 25 * 24 / 6 possibilities for unordered choices
3
Ok
`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?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your Solution:
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
3
Ok
`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?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
7 ways in the first bag to contain a ring and 6 in the second bag, 7*6/2= 21 possible combinations will contain 2 rings. 26 * 25 / 2 = 325 possible combinations which contain 2 tiles. 26 * 7 / 2 = 91 possible two-bag combinations containing a tile and a ring. 15 + 26 + 7 = 48 bags, so the total number of possible two-bag combinations is 48 * 47 / 2. 15 + 7 = 22 of the bags do not contain tiles, there are 22 * 21 / 2 two-bag combinations with no tiles. 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.
2
Ok
`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?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
15 * 26 * 25 * 7 * 14 ways.
3
Ok
`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?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
15 * 26 * 25 * 7 * 14 / ( 2 * 2)
3
Ok
*********************************************
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?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
22 * 21 * 20 * 19 * 18 / (5 * 4 * 3 * 2 * 1)
3
Ok
*********************************************
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, then how many 5-character 'words' consisting of 3 numbers followed by 2 letters could be formed from the results?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
15 * 15 * 15 * 26 * 26 possible 5-character 'words'
3
Ok
"
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