# mathematics

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In how many ways can the letters of MAHABHARAT be arranged so that
i) B n T are never together
ii) M,A,T n H occurs at first four places

• mathematics -

10 letters with 4 A's , 2 H's

a) without any restrictions, number of arrangements
= 10!/(4!2!) = 75600

consider the BT as one element
so we have 9 to arrange, with 4 A's and 2 H's
= 9!/(4!2!) = 7560

number of ways with the BT apart = 75600-7560 = 68040

b) Put the MATH in the front, that leaves
6 letters to arrange, containing of 3 A's
the remaining 6 letters can be arranged in 6!/3! or 120 ways.

That answer assumes that the front MATH stays that way.
If the MATH can be further arranged, but those letters still at the front,
then the number of ways would be 4!x(120) = 2880

• mathematics -

The scale of a map is given as 1:2,00000. Two cities are 3 cm apart on the map. Calculate the actual distance between them.

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