Some Results Related to nCr:

Illustration 1:

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

Solution:

##### 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

Ans.    (B)

Solution:

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

Ans.    (A)

Solution:

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.