Different point total are possible = 497
Step-by-step explanation:
Given - Four darts are thrown at this dartboard. Dartboard regions are
1, 4, 7, 10 points
To find - If all four darts hit the board, how many different point totals are
possible.
Proof -
Are given there are 4 regions -
1st region - 1 point
2nd region - 4 points
3rd region - 7 points
4th region - 10 points
Case I :
If all the darts hit the different region -
Total points possible are - 1 + 4 + 7 + 10 = 22 points
Case II :
If all the darts hit the same region -
It means either they hit 1 region or 2nd region or 3rd region or 4th region
If they all 4 hit first region , points are - 1+1+1+1 = 4
If they all 4 hit second region , points are - 4+4+4+4= 16
If they all 4 hit third region , points are - 7+7+7+7 = 28
If they all 4 hit fourth region , points are - 10+10+10+10 = 40
So,
the Total points possible are - 4 + 16 + 28 + 40 = 88
Case III :
If 3 darts hit the same region -
Sub-case 1 :
If 1 dart hit 1st region, other 3 dart hit 2nd region
Points are - 1 + 4 + 4+ 4 = 13
Sub-case 2 :
If 1 dart hit 1st region , other 3 dart hit 3rd region
Points are - 1 + 7 + 7 + 7 = 22
Sub-case 3 :
If 1 dart hit 1st region , other 3 dart hit 4th region
Points are - 1 + 10 + 10 + 10 = 31
Sub-case 4 :
If 1 dart hit 2nd region , other 3 dart hit 1st region
Points are - 4 + 1 + 1 + 1 = 7
Sub-case 5 :
If 1 dart hit 2nd region , other 3 dart hit 3rd region
Points are - 4 + 7 + 7 + 7 = 25
Sub-case 6 :
If 1 dart hit 2nd region , other 3 dart hit 4th region
Points are - 4 + 10 + 10 + 10 = 34
Sub-case 7 :
If 1 dart hit 3rd region , other 3 dart hit 1st region
Points are - 7 + 1 + 1 + 1 = 10
Sub-case 8 :
If 1 dart hit 3rd region , other 3 dart hit 2nd region
Points are - 7 + 4 + 4 + 4 = 19
Sub-case 9 :
If 1 dart hit 3rd region , other 3 dart hit 4th region
Points are - 7 + 10 + 10 + 10 = 37
Sub-case 10 :
If 1 dart hit 4th region , other 3 dart hit 1st region
Points are - 10 + 1 + 1 + 1 = 13
Sub-case 11 :
If 1 dart hit 4th region , other 3 dart hit 2nd region
Points are - 10 + 4 + 4 + 4 = 22
Sub-case 12 :
If 1 dart hit 4th region , other 3 dart hit 3rd region
Points are - 10 + 7 + 7 + 7 = 31
So,
Total points possible are - 13+22+21+7+25+34+10+19+37+13+22+31 = 254
Case IV :
If 2 darts hit the same region -
Sub-case 1:
If 2 darts hits 1st region , other 2 darts hit 2nd region
Points are - 1 + 1 + 4 + 4 = 10
Sub-case 2:
If 2 darts hits 1st region , other 2 darts hit 3rd region
Points are - 1 + 1 + 7 + 7 = 16
Sub-case 3:
If 2 darts hits 1st region , other 2 darts hit 4th region
Points are - 1 + 1 + 10 + 10 = 23
Sub-case 4:
If 2 darts hits 2nd region , other 2 darts hit 3rd region
Points are - 4 + 4 + 7 + 7 = 22
Sub-case 5:
If 2 darts hits 2nd region , other 2 darts hit 4th region
Points are - 4 + 4 + 10 + 10 = 28
Sub-case 6:
If 2 darts hits 3rd region , other 2 darts hit 4th region
Points are - 7 + 7 + 10 + 10 = 34
So,
Total points possible are - 10 + 16 + 23 + 22 + 28 + 34 = 133
Case V :
If 1 darts hit the same region -
This case is include in the Case III.
∴ we get
Different point total are possible = Points in Case I + II + III + IV
= 22 + 88 + 254 + 133
= 497
⇒Different point total are possible = 497
