In how many ways can you arrange 4 blocks of the same shape and size

Question

In how many ways can you arrange 4 blocks of the same shape and size in a row from left to right after selecting them from one orange, one green, one black, one purple, one white, two red, two yellow and two blue blocks?

(A) 2454
(B) 1950
(C) 990
(D) 975
(E) 960

(A) 2454 
(B) 1950 
(C) 990
(D) 975
(E) 960

Bunuel · Accepted Answer

So, we have eleven blocks of  eight  different colors: 
O; 
G; 
B; 
P;
W; 
R, R; 
Y, Y; 
L, L (let's denote blue as L).

We can have the following three cases:

1. All four blocks are of different colors, for example OGBP:   8C4*4! = 70*24 = 1680 . Here 8C4 gives the number of 4 different colors out of 8 and 4! is the number of arrangement of these 4 blocks;

2. Two blocks are of same color, and the remaining two are of the different colors, for example YYOG:   3C1*7C2*\frac{4!}{2!} = 3*21*12 = 756 . We have  three  colors with two blocks: RR, YY and LL. So, here 3C1 is the number of ways to choose the same color blocks: either RR, YY, or LL. 7C2 gives the number of ways to choose 2 remaining blocks out of 7 remaining colors. And 4!/2! is the number of arrangement of these 4 blocks out of which two blocks are of the same color;

3. Two blocks are of same color, and the remaining two are of the same color, for example YYLL:   3C2*\frac{4!}{2!2!} = 3*6= 18 . We have  three  colors with two blocks: RR, YY and LL. Here 3C2 is the number of ways to choose two out of three colors to give two blocks: either RRYY, RRLL, or YYLL. And 4!/(2!2!) is the number of arrangement of these 4 blocks.

Total = 1680 + 756 + 18 = 2454.

Answer: A.

Hope it helps.

ScottTargetTestPrep · Answer

Solution:

We see that there are 8 colors, of which 5 colors have a single block each and 3 colors have two blocks each. Therefore, the 4 blocks can be chosen from the following cases (note: in the calculations below: n in nCr means the number of colors available and r means the number of colors picked. Furthermore, the factor involving factorial means the number of ways one can arrange the 4 blocks once they are chosen. For example, 4! means the number of ways one can arrange the 4 blocks if they are all different colors and 4! / 2! means the number of ways one can arrange the 4 blocks if 2 of them have the same color):

1) All 4 blocks are of different colors:

8C4 x 4! = 70 x 24 = 1680

2) 2 blocks are of the same color and the other 2 are of different colors:

7C2 x 3C1 x 4!/2! = 21 x 3 x 12 = 756

3) 2 blocks are of one color and the other 2 are of the another color:

3C2 x 4!/(2!2!) = 3 x 6 = 18

Therefore, there are a total of 1680 + 756 + 18 = 2454 arrangements.

Answer: A

aniruddhawarang29 · Answer

Hello Scott,

In the following steps:

2) 2 blocks are of the same color and the other 2 are of different colors:
7C2 x 3C1 x 4!/2! = 21 x 3 x 12 = 756

Should it not be 7C2 x 3C 2  x 4!/2! = 21 x 3 x 12 = 756 --> Solution obtained is the same no doubt abt that.

RachitMehrotra · Answer

Bunuel  , can you publish a detailed explanation pls ?

StacyArko · Answer

Hi,
We select 1 out of the 3 same color block. Remember that 4!/2! accounts for the arrangement of the four blocks we selected.

Posted from my mobile device

bumpbot · Answer

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In how many ways can you arrange 4 blocks of the same shape and size

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