From 3 cocoa nuts, 4 apples, and 2 oranges how many how many selections of fruit can be made, taking at least one of each kind.
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Source: Beat The GMAT — Problem Solving |
The ways of choosing at least one cocoa=(2^3)-1=7maihuna wrote:From 3 cocoa nuts, 4 apples, and 2 oranges how many how many selections of fruit can be made, taking at least one of each kind.
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The ways of choosing at least one apple (2^4)-1=15
and for orange (2^2)-1=3
So total number of ways is 7*15*3=315
Another way of doing it, perhaps a little more complicated;
3C1 * 4c1 * 2c1 = 24
3c1 * 4c2 * 2c1 = 36
3c1 * 4c3 * 2c1 = 24
3c1 * 4c4 * 2c1 = 6
sum = 90
Now, for different choices of cocoa nuts, we can 3c1 or 3c2 or 3c3. 3c1 = 3c2, so we have 90 + 90 = 180. 3c3 = 1. so we have 90/3 = 30.
so we have 90+90+30 = 210. Then, we can have two oranges or 1 orange. for 2 oranges; 2c2 = 1, but 2c1 = 2. So, we have 210/2 = 105. So for all combinations with 2 oranges we have 105 possibilities. So total number ways = 210 + 105 = 315
3C1 * 4c1 * 2c1 = 24
3c1 * 4c2 * 2c1 = 36
3c1 * 4c3 * 2c1 = 24
3c1 * 4c4 * 2c1 = 6
sum = 90
Now, for different choices of cocoa nuts, we can 3c1 or 3c2 or 3c3. 3c1 = 3c2, so we have 90 + 90 = 180. 3c3 = 1. so we have 90/3 = 30.
so we have 90+90+30 = 210. Then, we can have two oranges or 1 orange. for 2 oranges; 2c2 = 1, but 2c1 = 2. So, we have 210/2 = 105. So for all combinations with 2 oranges we have 105 possibilities. So total number ways = 210 + 105 = 315
Hi,
Thank you for your explanation. What I'm not getting though is why did you use the number "2". I'm referring specifically to the "2"raised to the power of 3, then to the power of 4 and to the power of 2 in the following:
The ways of choosing at least one cocoa=(2^3)-1=7
The ways of choosing at least one apple (2^4)-1=15
and for orange (2^2)-1=3
So total number of ways is 7*15*3=315
My question is how did you come up with the 2 while the question is asking to choose one of each type of fruit.
I appreciate the clarification. Thank you.
Thank you for your explanation. What I'm not getting though is why did you use the number "2". I'm referring specifically to the "2"raised to the power of 3, then to the power of 4 and to the power of 2 in the following:
The ways of choosing at least one cocoa=(2^3)-1=7
The ways of choosing at least one apple (2^4)-1=15
and for orange (2^2)-1=3
So total number of ways is 7*15*3=315
My question is how did you come up with the 2 while the question is asking to choose one of each type of fruit.
I appreciate the clarification. Thank you.
you don't have do to 2..that's just a shortcut way of determining the possibilities; it would simpler to logically conclude that choosing at least 1 cocoa = 3c1 + 3c2 + 3c3 = 7
4c1 + 4c2 + 4c3 + 4c4 = 4 + 6 + 4 + 1 =15
2c1 + 2c2 = 3
7*15*3 = 315
This is actually simpler than the method I used, but the other one is a lot shorter in my opinion.
4c1 + 4c2 + 4c3 + 4c4 = 4 + 6 + 4 + 1 =15
2c1 + 2c2 = 3
7*15*3 = 315
This is actually simpler than the method I used, but the other one is a lot shorter in my opinion.
Lemme explain choosing one case, say cocoa, that will do the rest of the job.knowledgeisatreasure wrote:Hi,
Thank you for your explanation. What I'm not getting though is why did you use the number "2". I'm referring specifically to the "2"raised to the power of 3, then to the power of 4 and to the power of 2 in the following:
The ways of choosing at least one cocoa=(2^3)-1=7
The ways of choosing at least one apple (2^4)-1=15
and for orange (2^2)-1=3
So total number of ways is 7*15*3=315
My question is how did you come up with the 2 while the question is asking to choose one of each type of fruit.
I appreciate the clarification. Thank you.
Three cocoas are there.
U need to choose at least one cocoa. That means u can take one cocoa, or two cocoa or three cocoa. That means number of ways: 3C1+3C2+3C3
Now U need to know (for fast solving) there is a formula in general,
nC0+nC1+nC2+...............nCn=2^n
or nC1+nC2+...............nCn=2^n-nC0=2^n-1
Now please look at the above expression i.e. 3C1+3C2+3C3 and apply the above formula; we get 3C1+3C2+3C3 = (2^3)-1=7
similarly u can proceed for the other terms..
Should u need any further explanation please tell me.
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hariharakarthi
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What if a problem is like at least 2 of each kind.
From 3 cocoa nuts, 4 apples, and 2 oranges how many how many selections of fruit can be made, taking at least two of each kind.
Then the formula will be like 3C2+3C3
nC0+nC1+nC2+...............nCn=2^n
nC2+...............nCn=2^n-nC0=2^n-(nC1+nC0)
3C2+3C3 = 4
2^n-(nC1+nC0) = 2^3 -(3C1+3C0)
= 8 - (3+1)
= 4
Rule: Atleast one then it is 2^n - (nC0)
Irrespective of value of n, nC0 = 1
Atleast two then it is 2^n - (nC0+nC1)
Correct me if I am wrong.
regards,
hhk
From 3 cocoa nuts, 4 apples, and 2 oranges how many how many selections of fruit can be made, taking at least two of each kind.
Then the formula will be like 3C2+3C3
nC0+nC1+nC2+...............nCn=2^n
nC2+...............nCn=2^n-nC0=2^n-(nC1+nC0)
3C2+3C3 = 4
2^n-(nC1+nC0) = 2^3 -(3C1+3C0)
= 8 - (3+1)
= 4
Rule: Atleast one then it is 2^n - (nC0)
Irrespective of value of n, nC0 = 1
Atleast two then it is 2^n - (nC0+nC1)
Correct me if I am wrong.
regards,
hhk
For each fruit you can do 2 things that is either select or not select. Now since you have 3 cocoa nuts, for each cocoa nuts you can do 2 things that is either select or not select. Hence 2*2*2 = 2^3. Now you subtract 1 from this because there is a case where you can select none of the three cocoa nuts (this is possible in only one way) hence 2^3 -1.knowledgeisatreasure wrote:Hi,
Thank you for your explanation. What I'm not getting though is why did you use the number "2". I'm referring specifically to the "2"raised to the power of 3, then to the power of 4 and to the power of 2 in the following:
The ways of choosing at least one cocoa=(2^3)-1=7
The ways of choosing at least one apple (2^4)-1=15
and for orange (2^2)-1=3
So total number of ways is 7*15*3=315
My question is how did you come up with the 2 while the question is asking to choose one of each type of fruit.
I appreciate the clarification. Thank you.
Follow the similar logic to the rest of the fruits.
Thanks,
Quant-Master
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