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# A child throws six differently colored candies up in the air. How many

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Joined: 02 Sep 2009
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A child throws six differently colored candies up in the air. How many  [#permalink]

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13 May 2017, 14:24
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55% (hard)

Question Stats:

64% (01:59) correct 36% (02:09) wrong based on 128 sessions

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A child throws six differently colored candies up in the air. How many different possible groups of at least one candy are there that she could catch in her mouth?

A. 50
B. 51
C. 62
D. 63
E. 72

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A child throws six differently colored candies up in the air. How many  [#permalink]

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13 May 2017, 17:14
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The child catches: 1 candy, 2 candies, 3 candies, 4 candies, 5 candies or all 6 candies in her mouth.
This is a combination problem
She catches 1 candy = 6C1 different ways = 6 different ways
She catches 2 candies = 6C2 different ways
She catches 3, 4 , 5 or 6 = 6C3, 6C4, 6C5, 6C6 different ways.
Total possible ways = 6C1 + 6C2 + 6C3 + 6C4 + 6C5 +6C6
either solve all these one by one and add
OR
we know nC0 + nC1 + ...nCn = 2^n,
therefore nC1 +... nCn = 2^6 - 1 = 63 ways choice D
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Re: A child throws six differently colored candies up in the air. How many  [#permalink]

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13 May 2017, 21:46
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Bunuel wrote:
A child throws six differently colored candies up in the air. How many different possible groups of at least one candy are there that she could catch in her mouth?

A. 50
B. 51
C. 62
D. 63
E. 72

Total Candies = 6

When first candy is thrown in air then the possible outcomes = 2 (Either child catches in mouth or not)

When Second candy is thrown in air then the possible outcomes = 2 (Either child catches in mouth or not)

When Third candy is thrown in air then the possible outcomes = 2 (Either child catches in mouth or not)

When Forth candy is thrown in air then the possible outcomes = 2 (Either child catches in mouth or not)

When Fifth candy is thrown in air then the possible outcomes = 2 (Either child catches in mouth or not)

When Sixth candy is thrown in air then the possible outcomes = 2 (Either child catches in mouth or not)

Total Possible outcomes = 2x2x2x2x2x2 = 64

But these 64 outcomes include one case in which all the candies are missed from being caught in mouth so we need to remove it

Total Favourable cases = 64-1 = 63

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Re: A child throws six differently colored candies up in the air. How many  [#permalink]

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14 May 2017, 14:22
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The answer is a combination of all possible outcomes: one candy, two candies, three candies, four candies, five candies and six candies.

One Candy: 6C1 = 6

Two Candies: 6C2= $$\frac{6*5}{1*2}=15$$

Three Candies: 6C3= $$\frac{6*5*4}{1*2*3}=20$$

Four Candies: 6C4= $$\frac{6*5*4*3}{1*2*3*4}=15$$

Five Candies: 6C5= $$\frac{6*5*4*3*2}{1*2*3*4*5}=6$$

Six Candies: 6C6= $$\frac{6*5*4*3*2*1}{1*2*3*4*5*6}=1$$

Total: 6+15+20+15+6+1=63

The correction option is D
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Re: A child throws six differently colored candies up in the air. How many  [#permalink]

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15 May 2017, 21:16
case 1 = she catches 1 candy in her mouth = number of ways = 6C1 = 6
case 2 = number of ways of catching 2 candies in her mouth = 6C2 = 15
case 3= number of ways of catching 3 candies in her mouth = 6C3 = 20
case 4 = number of ways of catching 4 candies in her mouth = 6C4 = 15
case 5 = number of ways of catching 5 candies in her mouth = 6C5 = 6
case 6 = number of ways of catching all 6 candies in her mouth = 6C6 = 1

total = 63 Hence OPTION D
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Re: A child throws six differently colored candies up in the air. How many  [#permalink]

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18 May 2017, 20:06
Bunuel wrote:
A child throws six differently colored candies up in the air. How many different possible groups of at least one candy are there that she could catch in her mouth?

A. 50
B. 51
C. 62
D. 63
E. 72

Since the order of the candies within a group of candies doesn’t matter, this is a combination problem.

Number of ways the group of candies consists of exactly 1 candy: 6C1 = 6

Number of ways the group of candies consists of exactly 2 candies: 6C2 = (6 x 5)/2 = 15

Number of ways the group of candies consists of exactly 3 candies: 6C3 = (6 x 5 x 4)/(3 x 2) = 20

Number of ways the group of candies consists of exactly 4 candies: 6C4 = 6C2 = 15

Number of ways the group of candies consists of exactly 5 candies: 6C5 = 6C1 = 6

Number of ways the group of candies consists of all 6 candies: 6C6 = 1

Therefore the number of ways the group of candies consists of at least 1 candy is 6 + 15 + 20 + 15 + 6 + 1 = 63.

Alternative solution:

This is an “at least one” problem. Therefore, the number of ways the group of candies consists of at least 1 candy is the number of ways the group of candies consists of any number (from 0 to 6 inclusive) of candies, minus the number of ways the group of candies consists of exactly 0 candies.

The number of ways the group of candies consists of any number of candies is 2^6 = 64 and the number of ways the group of candies consists of exactly 0 candies is 6C0 = 1. Thus, the number of ways the group of candies consists of at least 1 candy is 64 - 1 = 63.

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Re: Quant PS Ques : Combinations  [#permalink]

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01 Jun 2017, 08:11
1
jasic wrote:
A child throws six differently colored candies up in the air.How many different possible groups of at least one candy are there that she could catch in the mouth

a) 50
b)31
c)64
d)63
e)721

Answer is D. It is the binomial expansion of 2^6. Since atleast one candy should fall in the mouth, we should exclude the one way in which no candies fall in her mouth. so naswer is 64-1 = 63
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Re: Quant PS Ques : Combinations  [#permalink]

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01 Jun 2017, 08:15
1
All cases - None will give answer...
Number of combinations = 64
-----> 64-1 = 63
D
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A child throws six differently colored candies up in the air. How many  [#permalink]

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01 Jun 2017, 09:47
Bunuel wrote:
jasic wrote:
A child throws six differently colored candies up in the air.How many different possible groups of at least one candy are there that she could catch in the mouth

a) 50
b)31
c)64
d)63
e)721

Merging topics. Please refer to the discussion above.

Answer is D. It is the binomial expansion of 2^6. Since atleast one candy should fall in the mouth, we should exclude the one way in which no candies fall in her mouth. so naswer is 64-1 = 63

Please give Kudos f you like my Post
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Re: A child throws six differently colored candies up in the air. How many  [#permalink]

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13 Jun 2017, 12:42
Total possible ways=
6C1+6C2+6C3+6C4+6C5+6C6

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Re: A child throws six differently colored candies up in the air. How many  [#permalink]

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27 Jun 2019, 02:00
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Re: A child throws six differently colored candies up in the air. How many   [#permalink] 27 Jun 2019, 02:00
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