In the above diagram is the centre of the larger circle. and are points on the circumference of the larger circle. and are points on the circumference of the smaller circle. Show that . and are straight lines.
(radii of the larger circle)
At a line from to (it is also a radius of the larger circle)
In combinatorial mathematics, a derangement is a permutation of the elements of a set in which no element appears in its original position.
For example, if we have the set , there are permutations
But only of them are derangements – and
I did a practical question on this here. In that question I used a tree diagram, but there must be a way to determine the number of derangements given elements.
We know elements has derangements, what about elements? We know there are permutations
ABCD
BACD
CABD
DABC
ACBD
BADC
CADB
DACB
ACDB
BCAD
CBAD
DBAC
ABDC
BCDA
CBDA
DBCA
ADBC
BDAC
CDAB
DCAB
ADCB
BDCA
CDBA
DCBA
I have highlighted the derangements. when there are derangements.
Let’s try to generalise.
In how many ways can one element be in its original position? Choose one element from the 4 to be in its original position, and then arrange the remaining three elements.
So far, we have
Clearly we are counting some arrangements multiple times, for example ABDC has A and B in the correct position, so we need to add all of the arrangements with elements in their original position.
In how many ways can two elements be in their original position?
So now we have,
But once again we have counted some arrangements multiple times, so we need to subtract all of the arrangements with elements in their original position.
In how many ways can three elements be in their original position?
So now we have,
We now need to add all of the arrangements with elements in their original position.
In how many ways can four elements be in their original position?
Four teachers decide to swap desks at work. How many ways can this be done if no teacher sits at their previous desk? Mathematics Specialist Units 1&2 Cambridge
I like this question as it seems easy until you start thinking about it. I think the best approach is a tree diagram.
If we think of the four teachers as A, B, C and D. Then A can no longer sit in A, so the options are B, C and D for the first desk.
For the second desk, If B is in the first desk, then A, C or D could be in the second. If C is in the first desk, then A or D could be in the second (B can’t be in the same desk). If D is in the first desk, then A or C can be in the second desk.
At 10am, object travelling with constant velocity km/h is sighted at the point with position vector km. At 11am object travelling with constant velocity km/h is sighted at the point with position vector km respectively. Use a scalar product method to determine given that the two objects were closest together at a distance of km at 4pm.
OT Lee Mathematics Specialist Year 11
At 4pm is at the point with position vector
and is at the point with position vector
We know the distance between and at 4pm is km.
Hence,
(1)
In the diagram below, I have found the position vector of relative to and the velocity of relative to
We know that when and are the closest distance apart.
(2)
Two equations and two unknowns which we can solve simultaneously. Both equations are circles.
Equation becomes
(3)
and equation becomes
(4)
From equation
We will worry about the negative version later.
Substitute for into equation
Square both sides of the equation
(5)
Substitute into
(6)
Substitute into
(7)
Now we need to consider the negative version of . If you work through (like I did above) you end with the same equation for .
Hence our two values for are or .
Would someone be expected to do this in an exam? I hope not, but I think its worth doing.