Problem Solving

Each of the three dice has its six faces numbered respectively 1,2,3,4,5,6 but the dice themselves are of different colours. if three dice are simultaneously thrown out of the dice box in how many of these way they fall in such a way that two of the dice show same number and the third a different number.

90
120
210
216
1296

is it 90?

Since I did guess the previous permutations question I attempted, I'll try this one too.

First off let's not think of the dice as colored. Consider three identical dice and you're trying to find out the ways in which two hit the same number while the third hits a different one. There are 6 numbers that the two dice can show together: 1 through 6. For each of the 6 possibilities, the third dice has to show 5 other numbers (EXCEPT the one shown by the other two). In total, you have 6*5 = 30 possibilities.

But the dice are also colored, which adds an extra twist: in this case, there will be 3C2 = 3 possible combinations of the two "common" dice. The third dice doesn't matter now, since it will be the one left.

This makes my guess 30*3 = 90.

I agree. Derived 90 via same approach.

What is the OA?

Cheers.

DanaJ wrote:Since I did guess the previous permutations question I attempted, I'll try this one too.

First off let's not think of the dice as colored. Consider three identical dice and you're trying to find out the ways in which two hit the same number while the third hits a different one. There are 6 numbers that the two dice can show together: 1 through 6. For each of the 6 possibilities, the third dice has to show 5 other numbers (EXCEPT the one shown by the other two). In total, you have 6*5 = 30 possibilities.

But the dice are also colored, which adds an extra twist: in this case, there will be 3C2 = 3 possible combinations of the two "common" dice. The third dice doesn't matter now, since it will be the one left.

This makes my guess 30*3 = 90.

OA is 90 but I m yet not able to comprehend the logic

Read this thread..it may help...focus on the part B of the solution...

https://www.beatthegmat.com/dice-t42198. ... light=dice

Cheers.

maihuna wrote:OA is 90 but I m yet not able to comprehend the logic

Blue.............Red....................Pink
6 poss..........1 poss................5 poss. = 6 * 5 * 1 = 30

5 poss..........6 poss................1 poss. = 5 * 6 * 1 = 30

1 poss..........5 poss................6 poss. = 1 * 5 * 6 = 30

30 + 30 + 30 = 90

Think of it this way; each color will either have 5 possibilities or 1 possibility or 6 possibilities., so you should account for all three scenarios

maihuna,

I got 90 by this logic(no need to think about the colour at all)

there are 5 different possibilities of getting same number twice

(1,1,x) (2,2,x)........(6,6,x)

Number of different permutations of (1,1,x) is 3!/2!

formula used : number of different arrangements of n items with r similar items is (n!/r!)

x can take 5 different values - so (3!/2!) * 5 = 15

for (1,1,x) it is 15..

for all the six possibilities it is 6 * 15 = 90

dont confuse by colours......because anyway (1,2) is different from (2,1) - even if u have the same colour dice.