Question

The number of ways in which we can get a score of 11 by throwing three dice is...

Answer 1

HI!!

Answer 2

you familiar with dice?

Answer 3

yeah i am..

Answer 4

ok i guess we got a bunch of counting to do then because you have 3 dice total, not the usual 2

Answer 5

yep two dices would be easy.. but there's 3

Answer 6

none of the dice can show a one, but if one shows a 2, then it can be
\[(2,5,6)\] or any combination of those
there are 6 i believe \[(2,5,6),(2,6,5),(5,2,6),(5,6,2),(6,2,5),(6,5,2)\] or \(3!=6\) ways where one is a two

Answer 7

18,27,45,56 are the options

Answer 8

now lets move on to one showing a 3
\[(3,2,6)\] 6 of those
\[(3,3,5)\] 3 of those
\[(3,4,4)\] 3 of those
guess that is it for one of them showing a 3

Answer 9

ready to move on to 4? we have to make sure we don't count anything twice

Answer 10

oh i made a mistake
one could show a 1 lets count those too

Answer 11

\[(1,5,5)\] 3 of those and
\[(1,4,6)\] 6 of those

Answer 12

keep counting in this way, you will get it

Answer 13

oh nvm we are done
just count them up

Answer 14

you made 12 cases

Answer 15

for two dice the pattern is
\[1,2,3,4,5,6,5,4,3,2,1\]

Answer 16

yeah count the number of ways for each
for 3 dice the pattern is
\[1,3,6,10,15,21,25,27,27,25,21,15,10,6,3,1\]

Answer 17

for rolling respectively
\[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18\]

Answer 18

notice that
\[(1,4,6)\] where all the numbers are different has \(3!=6\) possible ways to do it

Answer 19

its confusing in counting cases

Answer 20

whereas \((1,5,5)\) has only 3 possible ways
\[(1,5,5),(5,1,5),(5,5,1)\]

Answer 21

ok lets go slow

Answer 22

you got one possibility is \((1,4,6)\) right because they add up to 11

Answer 23

but you have \(3!=6\) permutations of those three numbers

Answer 24

6 is not so big that we cannot list them
\[(1,4,6),(1,6,4),(4,1,6),(4,6,1),(6,1,4),(6,4,1)\] six ways to do it

Answer 25

ok...

Answer 26

so lets do them in order

Answer 27

\[(1,4,6)\] 6 choices
\[(1,5,5)\] 3 choices

Answer 28

\[(2,3,6)\] 6 choices
\[(2,4,5)\] 6 choices

Answer 29

\[(3,3,5)\]3 choices
\[(3,4,6)\] 6 choices