Goldenrod and No Hope are in a horse race
with 6 contestants. How many different
arrangements of finishes are there if No Hope
always finishes before Goldenrod and if all of the
horses finish the race?
(A) 720
(B) 360
(C) 120
(D) 24
(E) 21
Solution:
_ _ _ _ _ _ (6 dashes to be equated to 6 positions)
Case 1: No hope comes first, then number of arrangements with Goldenrod coming second is 5!
Case 2: No hope comes second, , then number of arrangements with Goldenrod coming second is 4!
Case 3 :No hope comes third then number of arrangements with Goldenrod coming second is 3!
Case 4 :No hope comes fourth, , then number of arrangements with Goldenrod coming second is 2!
Case 5 :No hope comes fifth, , then number of arrangements with Goldenrod coming second is 1 way
5! + 4! + 3! +2! +1 = 153, but answer is 360.
Can anyone tell me why my solution is wrong. What have i missed calculating??
Thanks
horses and positions - Tough math
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- Morgoth
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Case 1: No hope comes first, then number of arrangements with Goldenrod coming second is 5 * 4!gmatrant wrote:
Solution:
_ _ _ _ _ _ (6 dashes to be equated to 6 positions)
Case 1: No hope comes first, then number of arrangements with Goldenrod coming second is 5!
Case 2: No hope comes second, , then number of arrangements with Goldenrod coming second is 4!
Case 3 :No hope comes third then number of arrangements with Goldenrod coming second is 3!
Case 4 :No hope comes fourth, , then number of arrangements with Goldenrod coming second is 2!
Case 5 :No hope comes fifth, , then number of arrangements with Goldenrod coming second is 1 way
5! + 4! + 3! +2! +1 = 153, but answer is 360.
Thanks
Case 2: No hope comes second, , then number of arrangements with Goldenrod coming second is 4* 4!
Case 3 :No hope comes third then number of arrangements with Goldenrod coming second is 3*4!
Case 4 :No hope comes fourth, , then number of arrangements with Goldenrod coming second is 2*4!
Case 5 :No hope comes fifth, , then number of arrangements with Goldenrod coming second is 4!
5*4! + 4*4! + 3*4! + 2*4! + 4! = 4! * 15 = 24 * 15 = 360
Hope this helps.
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If you were to draw slots it would look like this:
NH - No Hope (any slot wiht NH has value 1 since we are fixing its spot)
Case1:
NH COMES FIRST
NH 5 4 3 2 1 = 120
---- ---- ---- ---- ---- ----
Case2:
NH COMES SECOND
4 NH 4 3 2 1 = 96
---- ---- ---- ---- ---- ----
Case3:
NH COMES THIRD
4 3 NH 3 2 1 = 72
---- ---- ---- ---- ---- ----
Case4:
NH COMES FOURTH
4 3 2 NH 2 1 = 48
---- ---- ---- ---- ---- ----
Case5:
NH COMES FIFTH
4 3 2 1 NH 1 = 24
---- ---- ---- ---- ---- ----
Add all you would get 360
Hope this helps!
NH - No Hope (any slot wiht NH has value 1 since we are fixing its spot)
Case1:
NH COMES FIRST
NH 5 4 3 2 1 = 120
---- ---- ---- ---- ---- ----
Case2:
NH COMES SECOND
4 NH 4 3 2 1 = 96
---- ---- ---- ---- ---- ----
Case3:
NH COMES THIRD
4 3 NH 3 2 1 = 72
---- ---- ---- ---- ---- ----
Case4:
NH COMES FOURTH
4 3 2 NH 2 1 = 48
---- ---- ---- ---- ---- ----
Case5:
NH COMES FIFTH
4 3 2 1 NH 1 = 24
---- ---- ---- ---- ---- ----
Add all you would get 360
Hope this helps!