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1) A jar contains four coins: a nickel, a dime, a quarter, and a half-dollar. Three coins are randomly selected from the jar. The sample space therefore consists of {NDQ, NDH, NQH, DQH}. Each of the four simple outcomes is equally likely. Find the following probabilities P(the value of the three coins selected totals at least 80cents) P(the value of the three coins selected totals exactly 65 cents)
In how many ways can 19 coins equal exactly one dollar?
Discussion/Solution? :
What are the possible types of coins that can be used to obtain a sum of one dollar?penny, nickel, dime, quarter, half dollar
So there are five different types of coins.
Strategy:
Consider five cases involving
(1) only one type of coin
(2) only two types of coins
(3) only three types of coins
(4) only four types of coins
(5) all five types of coins
Solutions:
Case (1): It is impossible to have 19 coins of the same type that equal one dollar.
19 pennies = 19 cents
19 nickels = 95 cents
19 dimes = $1.90 (too big so quarters and half dollars are also eliminated)
Case (2): There are TWO ways to have one dollar using only two types of coins.
10 pennies and 9 dimes = $1
18 nickels and 1 dime = $1
Case (3): There are THREE ways to have one dollar using only three types of coins.
15 pennies, 1 dime, and 3 quarters = $1
10 pennies, 8 nickels, and 1 half dollar = $1
5 pennies, 9 nickels, and 5 dimes = $1
Case (4): There are FOUR ways to have one dollar using only four types of coins.
15 pennies, 2 nickels, 1 quarter, and 1 half dollar = $1
10 pennies, 3 nickels, 5 dimes, and 1 quarter = $1
10 pennies, 6 nickels, 1 dime, and 2 quarters = $1
5 pennies, 12 nickels, 1 dime, and 1 quarter = $1
Case (5): It is impossible to have all five types of coins.
Suppose we did have all five types of coins. We could have at most one half dollar. So we have accounted for 50 cents. That leaves 50 cents for the other four types of coins. If we are to have all four types then we can have at most one quarter (25 cents). We must now split the remaining 25 cents among the dimes, nickels, and pennies. If we have just one dime, one nickel, and one penny (16 cents), then the remaining 9 cents must be in pennies (i.e. 9 pennies) or made up of another nickel and 4 more pennies. We now have
10 pennies, 1 nickel, 1 dime, 1 quarter, and 1 half dollar = $1
or 5 pennies, 2 nickels, 1 dime, 1 quarter, and 1 half dollar = $1
However, we only have 10 + 1 + 1 + 1 + 1 = 14 coins or 5 + 2 + 1 + 1 + 1 = 10 coins
( not the 19 coins we wanted).
Therefore, it is impossible to use all five types of coins to get 19 coins to equal one dollar.
Observations:
1. The number of half dollars must be 0, 1.
2. The number of quarters must be 0, 1, 2, 3.
3. The number of dimes must be 0, 1, 5, 9.
4. The number of nickels must be 0, 2, 3, 6, 8, 9, 12, 18.
5. The number of pennies must be 0, 5, 10, or 15.
The first two observations are obvious. Since half dollars are worth 50 cents each, we can have at most one. Also, since quarters are worth 25 cents each, we can have at most 3.
What about the other observations? Obviously, we can have at most 9 dimes, but why only 0, 1, 5, or 9? Why not 2, 3, 4, 6, 7, or 8? It may be easier to consider the pennies first.
Let p, n, d, q, hd represent the number of pennies, nickels, dimes, quarter, and half dollars, respectively.
So
p + 5n + 10d + 25q + 50 hd = 100
If we evaluate the equation (mod 5) we get
p = 0 (mod 5)
i.e. p = 0, 5, 10, 15, ... but since we can only have 19 coins, there is at most 15 pennies
Therefore, p = 0, 5, 10, or 15. i.e. p = 5k where k = 0,1,2,3
Return to Susie Lanier's Class Page.
Video transcript
- Let's get some practice counting money! So I have six coins right over here, and these are all United States coins, we're counting money in the United States for these examples, and what is this first coin? Well, this is called a quarter, or a quarter-dollar, so it represents 25 cents, and we could write it out as 25 cents, but I'll just keep that one like that. Now this one's another quarter, so it is also going to be 25 cents. Now this one looks different, but it's just the other side of these coins. This is what the other side looks like. So this is also going to be 25 cents. These are three quarters right over here. So how much money do these three quarters represent? Well, it's going to be 25 plus 25, which is 50, plus 25, which is going to be 75, so these three quarters are going to be 75 cents. And remember, 100 cents make a dollar, so this is still less than one dollar. But we're not done yet. We have this nickel, this is a nickel right over here, that represents five cents, and then we have another nickel here, it looks different, but it's just the other side, this is the head side, this is the tail side. So this is also another five cents, and so these two nickels, if you add them together, they are going to represent 10 cents, and then finally, you have a penny, and a penny, and it even says it right over here, is one cent, in fact, they all say it here, this is five cents, this is one cent. So this right over here's gonna be one cent. So what is 75 plus 10 plus 1? Well 75 plus 10 is 85, plus 1 is 86, so this is equal to 86 cents. And if it felt a little bit too fast to count up 25, 50, 75 in your head, you could also add them up. 25, 25, 25, plus 5, another 5, plus 1. You could add them up this way, and then what would you get? 5 plus 5 is 10, plus 5 is 15, plus 5 is 20, plus 5 is 25, plus 1 is 26, so that's two tens and one six, put the two tens up here. 2 plus 2 is 4, 4 plus 2 is 6, 6 plus 2 is 8. So you could get 86 cents, either way. Let's do one that has even more coins in it. So here we go, we have, so what is it, what's going on here? This right over here is a quarter, that's going to be 25 cents. 25 cents, and then, we have one that we didn't see in the previous example, we have two dimes. A dime represents 10 cents, so we have two dimes, where we can have those each represent 10 cents, then we have two nickels, we've already seen those each represent five cents, so 5 and 5, and then we have four pennies, one, two, three, four. Now we could put each penny separately, like that, or we could say, look, four pennies, each of them represent one cent, so that's going to be four cents. So let me do it that way. So, this one, two, three, four, that's going to be four cents. And then we could just add everything up, so 5 plus 0 plus 0 plus 5 is 10, plus 5 is 15, plus 4 is 19. 19 is one ten and nine ones, so I could put the one ten in the tens place, 1 plus 2 is 3, plus 1 is 4, plus 1 is 5. So it's five tens, five tens and nine ones, so 59 cents. This right over here is 59 cents. And notice, we have more coins, but it represents less value than the previous example. That's because we had a lot of coins that didn't represent a lot of values, like, we had these four pennies here, while the previous example, we had three quarters! Each of these quarters is equivalent to 25 pennies, so we're able to represent more money with fewer coins in the first example.
What coins make 80 cents?
A | B |
7 dimes + 2 nickels | 80 cents |
1 nickel + 3 pennies | 8 cents |
2 nickels + 2 pennies | 12 cents |
4 nickels + 9 pennies | 29 cents |