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
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
To find the number of ways to arrange the letters of the word "MAHABHARAT" satisfying the given conditions, we can use the principles of permutation and combination.
i) B and T are never together:
In order to calculate the arrangements where B and T are never together, we can consider them as a single entity. Let's call this entity as X, which includes both B and T. Now, we have the letters MAHAXHARAX. The total number of letters remaining is 8 (M, A, H, A, X, H, A, R, A), including the entity X.
The number of ways to arrange 8 letters (including repeats) is 8!/(2! * 3! * 2!). This is calculated by dividing the factorial of the total number of letters by the factorials of the number of repetitions of each letter: 2 repetitions of A, 3 repetitions of H, and 2 repetitions of X.
ii) M, A, T, and H occurs at the first four places:
Since M, A, T, and H should occur at the first four places, we have to arrange M, A, T, and H in these places. The number of ways to arrange these four letters is 4!.
Now, for the remaining letters (A, H, A, R, A, X), the number of ways to arrange them is 6!/ (3! * 2!), using the same logic as in condition i.
To find the final result, we need to multiply the number of arrangements satisfying both conditions i and ii:
Total Number of arrangements = Number of ways where B and T are never together * Number of ways M, A, T, and H occurs at first four places
Total Number of arrangements = (8!/(2! * 3! * 2!)) * (4! * 6!/(3! * 2!))
Now, we can calculate the final result.
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.
Scale of map = 1:200000
Distance between two cities = 3cms
Consider map scale = 1cm : 200000 cms
200000 cms = 2 Km
3 cms = 3 X 2 kms = 6 kms
Therefore, Actual distance between two cities = 6 kms.