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12 Mar 2021, 03:16
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A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
36% (02:59) correct
64%
(02:35)
wrong
based on 107
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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
13 Apr 2021, 10:40
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Bunuel
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.
General Discussion
29 Mar 2021, 11:06
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Bunuel
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
