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805+ (Hard)|   Combinations|      
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Yogananda
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Seven boxes numbered 1 to 7 are arranged in a row.Each is to be filled 17 May 2020, 00:44
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C
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95% (hard)

Question Stats:

44% (02:42) correct 56% (02:26) wrong based on 88 sessions

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Seven boxes numbered 1 to 7 are arranged in a row.Each is to be filled by either a black or blue colored ball such that no two adjacent boxes contain blue colored balls.In how many ways can the boxes be filled with the balls?
A.23
B.33
C.34
D.32
E.24
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C
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Seven boxes numbered 1 to 7 are arranged in a row.Each is to be filled 17 May 2020, 22:41
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Seven boxes numbered 1 to 7 are arranged in a row.Each is to be filled by either a black or blue colored ball such that no two adjacent boxes contain blue colored balls.In how many ways can the boxes be filled with the balls?
A.23
B.33
C.34
D.32
E.24
Show more


As there is a restriction on the placement of Blue balls, let us concentrate on the ways to pick blue balls in different circumstances.
Let us denote blue by B and black by X.

Case I : 0 blue balls,


So 7 Xs and 0B
_X_X_X_X_X_X_X_
The 0 blue ball can be placed in any of the gaps shown by _, => 8C0=1

Case II : 1 blue ball


So 6 Xs and 1B
_X_X_X_X_X_X_
The 1 blue ball can be placed in any of the gaps shown by _, => 7C1=7

Case III : 2 blue ball


So 5 Xs and 2 Bs
_X_X_X_X_X_
The 2 blue ball can be placed in any of the gaps shown by _, => 6C2=15

Case IV : 3 blue ball


So 4 Xs and 3 Bs
_X_X_X_X_
The 3 blue ball can be placed in any of the gaps shown by _, => 5C3=10

Case V : 4 blue ball


So 3 Xs and 4 Bs
_X_X_X_
The 4 blue ball can be placed in any of the gaps shown by _, => 4C4=1

Beyong this, the number of blue balls will be more than the gaps available so 0 ways.

Total = \(1+7+15+10+1=34\)

C
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Seven boxes numbered 1 to 7 are arranged in a row.Each is to be filled 17 May 2020, 00:55
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MBADream786
Seven boxes numbered 1 to 7 are arranged in a row.Each is to be filled by either a black or blue colored ball such that no two adjacent boxes contain blue colored balls.In how many ways can the boxes be filled with the balls?
A.23
B.33
C.34
D.32
E.24
Show more

Case 0: 0 Blue and 7 Black boxes can be arranged in = 1 ways

Case 1: 1 Blue and 6 Black boxes can be arranged in 7C1 (choose one out of 7 blue places) = 7!/6! = 7 ways
_ _ _ _ _ _ _ _ _ _ _ _ _

Case 2: 2 Blue and 5 Black boxes can be arranged in = 6C2 (choose 2 out of 6 blue places) = 15 ways
_ _ _ _ _ _ _ _ _ _ _

Case 3: 3 Blue and 4 Black boxes can be arranged in = 5C3 ways = 10 ways

Case 4: 4 Blue and 3 Black boxes can be arranged in = 1 ways

Total Possible ways = 1+7+15+10+1 = 34

Answer: Option C
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Seven boxes numbered 1 to 7 are arranged in a row.Each is to be filled 17 May 2020, 09:56
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Tricky way

suppose we have to fill first n-2 boxes and number of ways [f(n-2)]to fill them is x+y, where x ways ended up with Black and y ways ended up with Blue.

Then number of ways[f(n-1)] to fill n-1 boxes = 2x+y, where x+y ways ended up with Black and x ways ended up with Blue. (Why?)
As number of ways to fill subsequent box of black is 2(blue or black), and number of ways to fill subsequent box of blue is 1(black).

Then number of ways[f(n)] to fill n boxes = 2(x+y)+x= 3x+2y, where (2x+y) ways ended up with Black and (x+y) ways ended up with Blue.

We can see that f(n) = f(n-1)+f(n-2)

In our question. number of ways to fill first box = 1+1=2 (x=1, y=1) Either blue or black

number of ways to fill first 2 boxes = 2*1+1 = 3

Now it follows the above pattern - f(n) = f(n-1)+f(n-2)

f(3) = 2+3=5;
f(4)= 3+5=8
f(5)= 5+8=13
f(6)= 8+13=21
f(7)= 13+21=34 (Answer)

MBADream786
Seven boxes numbered 1 to 7 are arranged in a row.Each is to be filled by either a black or blue colored ball such that no two adjacent boxes contain blue colored balls.In how many ways can the boxes be filled with the balls?
A.23
B.33
C.34
D.32
E.24
Show more
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Seven boxes numbered 1 to 7 are arranged in a row.Each is to be filled 17 May 2020, 22:03
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MBADream786
Seven boxes numbered 1 to 7 are arranged in a row.Each is to be filled by either a black or blue colored ball such that no two adjacent boxes contain blue colored balls.In how many ways can the boxes be filled with the balls?
A.23
B.33
C.34
D.32
E.24

Case 0: 0 Blue and 7 Black boxes can be arranged in = 1 ways

Case 1: 1 Blue and 6 Black boxes can be arranged in 7C1 (choose one out of 7 blue places) = 7!/6! = 7 ways
_ _ _ _ _ _ _ _ _ _ _ _ _

Case 2: 2 Blue and 5 Black boxes can be arranged in = 6C2 (choose 2 out of 6 blue places) = 15 ways
_ _ _ _ _ _ _ _ _ _ _

Case 3: 3 Blue and 4 Black boxes can be arranged in = 5C3 ways = 10 ways

Case 4: 4 Blue and 3 Black boxes can be arranged in = 1 ways

Total Possible ways = 1+7+15+10+1 = 34

Answer: Option C
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Could you elaborate case 2 and 3 a little further. Whats the idea behind?

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Seven boxes numbered 1 to 7 are arranged in a row.Each is to be filled 17 May 2020, 22:11
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robinbjoern
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MBADream786
Seven boxes numbered 1 to 7 are arranged in a row.Each is to be filled by either a black or blue colored ball such that no two adjacent boxes contain blue colored balls.In how many ways can the boxes be filled with the balls?
A.23
B.33
C.34
D.32
E.24

Case 0: 0 Blue and 7 Black boxes can be arranged in = 1 ways

Case 1: 1 Blue and 6 Black boxes can be arranged in 7C1 (choose one out of 7 blue places) = 7!/6! = 7 ways
_ _ _ _ _ _ _ _ _ _ _ _ _

Case 2: 2 Blue and 5 Black boxes can be arranged in = 6C2 (choose 2 out of 6 blue places) = 15 ways
_ _ _ _ _ _ _ _ _ _ _

Case 3: 3 Blue and 4 Black boxes can be arranged in = 5C3 ways = 10 ways

Case 4: 4 Blue and 3 Black boxes can be arranged in = 1 ways

Total Possible ways = 1+7+15+10+1 = 34

Answer: Option C

Could you elaborate case 2 and 3 a little further. Whats the idea behind?

Posted from my mobile device
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robinbjoern

Case 2 is when we need to arrange 2 Blue and 5 Black boxes

Let's place 5 black boxes with some gaps between adjacent black boxes

_ _ _ _ _

Now we have 6 spaces left for 2 blue boxes as mentioned below

B _ B. _ B _ B _ B _ B

We need to fill 2 of these 6 spaces available for Blue so we choose 2 places out of 6 in 6C2 ways =n 15


We repeat the same exercise in case 3 as well
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Seven boxes numbered 1 to 7 are arranged in a row.Each is to be filled 17 May 2020, 23:07
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robinbjoern
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MBADream786
Seven boxes numbered 1 to 7 are arranged in a row.Each is to be filled by either a black or blue colored ball such that no two adjacent boxes contain blue colored balls.In how many ways can the boxes be filled with the balls?
A.23
B.33
C.34
D.32
E.24

Case 0: 0 Blue and 7 Black boxes can be arranged in = 1 ways

Case 1: 1 Blue and 6 Black boxes can be arranged in 7C1 (choose one out of 7 blue places) = 7!/6! = 7 ways
_ _ _ _ _ _ _ _ _ _ _ _ _

Case 2: 2 Blue and 5 Black boxes can be arranged in = 6C2 (choose 2 out of 6 blue places) = 15 ways
_ _ _ _ _ _ _ _ _ _ _

Case 3: 3 Blue and 4 Black boxes can be arranged in = 5C3 ways = 10 ways

Case 4: 4 Blue and 3 Black boxes can be arranged in = 1 ways

Total Possible ways = 1+7+15+10+1 = 34

Answer: Option C

Could you elaborate case 2 and 3 a little further. Whats the idea behind?

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robinbjoern

Case 2 is when we need to arrange 2 Blue and 5 Black boxes

Let's place 5 black boxes with some gaps between adjacent black boxes

_ _ _ _ _

Now we have 6 spaces left for 2 blue boxes as mentioned below

B _ B. _ B _ B _ B _ B

We need to fill 2 of these 6 spaces available for Blue so we choose 2 places out of 6 in 6C2 ways =n 15


We repeat the same exercise in case 3 as well
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Great way to look at it! Thanks for your time :)
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Seven boxes numbered 1 to 7 are arranged in a row.Each is to be filled 19 Feb 2025, 10:24
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