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## Quiz of the Month (April 2001)

### Hector C. Parr

***

#### SOLUTION TO LAST MONTH'S QUIZ

```         1) 20 mph

2) 37, 51, 60, 65, 87

3) (a) 9    (b) 265

Notes

1). Suppose normal speed = v mph
Then normal time = 5/v hrs.

So  5/(v + 10) = 5/v - 1/12

Solving,  v = 20  (or v = -30   ! )

2). By intelligent trial and error.

3). The following is a general solution:
Let f(n,m) represent the number of arrangements of
the numbers 1 to n with exactly m in their correct
positions. It is easy to show that
f(n,m) = nCm f(n-m,0) (where nCm is the number of
selections of n objects takem m at a time).

This enables the following table to be built one
column at a time, using also the fact that the total
for each column must be factorial n.

n  = 1    2    3    4    5    6    (etc)
m =  0    0    1    2    9    44   265
1    1    0    3    8    45   264
2         1    0    6    20   135
3              1    0    10    40
4                   1     0    15
5                         1     0
6                               1

Alternatively, solution by computer is possible.
```

#### THIS MONTH'S QUIZ

```  1. Two oranges and three lemons cost 119 pence. If an
orange costs 7p more than a lemon, what does a
lemon cost?

2. Andrew and Barry together have £29 in their pockets,
Barry and Charles together have £40, while Charles
and David together have £65. David has twice as much
as Andrew. How much has Barry?

3. A wine-glass contains 10 cl of wine, and a water-jug
contains 100 cl of water. If 1 cl of water is taken
from the jug and well mixed with the wine, and then
1 cl of the mixture is returned to the water-jug, which
is greater, the volume of water in the wine or wine in
the water?
```

***

(c) Hector C. Parr (2001)

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