Solved: Eight runners compete in a race where there are no tied finishes. Maria and Tom... [IB Mathematics Analysis and Approaches (AA)]

Question

HLPaper 2

Eight runners compete in a race where there are no tied finishes. Maria and Tom are two of the eight competitors in this race. Find the total number of possible ways in which the eight runners can finish if Tom finishes

in the position immediately after Maria.

[2]

Verified

Solution

Tom and Maria finish in that order (as a unit) so we are considering the arrangement of $7$ objects (M1)

$7! = 5040$ ways A1

[2 marks]

in any position after Maria.

[3]

Verified

Solution

METHOD 1

the number of ways that Maria finishes in front of Tom is equal to the number of ways that Tom finishes in front of Maria (M1)

total number of ways is ${8!}$ (A1)

ways = ${\frac{8!}{2} = 20160}$ A1

METHOD 2

the other six runners can finish in ${6! = 720}$ ways (A1)

when Maria finishes first, Tom can finish in ${7}$ different positions

when Maria finishes second, Tom can finish in ${6}$ different positions etc

ways = ${7+6+5+4+3+2+1 = 28}$ ways (A1)

hence there are ${(7+6+5+4+3+2+1) \times 6!}$ ways

ways = ${28 \times 6! = 20160}$ ways A1

[3 marks]

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