The total number of words (with or without meaning) that can be formed out of the letters of the word 'DISTRIBUTION' taken four at a time, is equal to _______. [2024]
(3734)
The word 'DISTRIBUTION' has total 12 letters and 9 diferent letters.
We select four different letters.
Case l : Three same letters and 1 different letter
Case 2 :Two pairs are same and two pairs are different
Case 3 : Two pair of letters are same
Case 4 : All different letters are select
Total letters selected.
There are 5 points on the side AB, excluding A and B, of a triangle ABC. Similarly, there are 6 points on the side BC and 7 points on the side CA of the triangle. The number of triangles, that can be formed using the points as vertices, is: [2024]
796
751
771
776
(2)
Number of points on side AB = 5
Number of points on side BC = 6
Number of points on side AC = 7
Number of ways of selecting three points from side AB =
Number of ways of selecting three points from side BC =
Number of ways of selecting three points from side AC =
Total number of triangles that can be formed using the points
The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is: [2024]
16
48
24
56
(1)
Total number of triangles that can be formed by using 8 vertices of the octagon =
Number of triangles having exactly one side common with the octagon =
Since, if we choose AB as the common side, then other vertices of the triangle will be either of G, F, E or D as we have exactly one common side.
So, we have i.e., such triangles.
Now, let us find the number of triangles having two common sides.
In the octagon, we have 8 ways of choosing two consecutive sides, i.e., (AB, BC), (BC, CD), (CD, DE), (DE, EF), (EF, FG), (FG, GH), (GH, HA), (HA, AB)
Number of triangles having 2 sides common with the octagon = 8
Required number of triangles =
If for some and then is equal to [2024]
372
384
380
376
(1)
Given,
i.e.,
or i.e., i.e.,