Some Results Related to nCr:
a) How many diagonals are there in an n-sided polygon (n> 3).
b) How many triangles can be formed by joining the vertices of an n- sided polygon?
How many of these triangles have:
(i) Exactly one side common with that of the polygon
(ii) Exactly two sides common with that of the polygon
(iii) No sides common with that of the polygon
ALL POSSIBLE SELECTIONS
Selection from Distinct Objects:
The number of selections from n different objects, taken at least one
= nC1 + nC2 + nC3 + —— + nCn = 2n – 1.
Question 1. In a chess tournament, all participants were to play one game with the other. Two players fell ill after having played 3 games each. If total number of games played in the tournament is equal to 84, then total number of participants in the beginning was equal to:
(A) 10 (B) 15
(C) 12 (D) 14
Let there were ‘n’ players in the beginning. Total number of games to be played was played to nC2 and each player would have played (n – 1) games. Thus
nC2 – ((n – 1) + (n – 1) – 1) + 6 = 84
n2 – 5n – 150 = 0
n = 15
Question 3. If letters of the word ‘KUBER’ are written in all possible orders and arranged as in a dictionary, then rank of the word ‘KUBER’ will be:
(A) 67 (B) 68
(C) 65 (D) 69
Alphabetical order of these letters is B, E, K, R, U.
Total words starting with B = 4! = 24
Total words starting with E = 4! = 24
Total words starting with KB = 3! = 6
Total words starting with KE = 3! = 6
Total words starting with K = 3! = 6
Next word will be KUBER.
Thus rank of the word KUBER = 24 + 24 + 18 + 1 = 67.
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