**Some Results Related to ^{n}C_{r}:**

*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

= ^{n}C_{1} +^{ n}C_{2 }+ ^{ n}C_{3 }+ —— +^{ n}C_{n} = 2^{n }– 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 ^{n}C_{2} and each player would have played (n – 1) games. Thus

^{n}C_{2} – ((n – 1) + (n – 1) – 1) + 6 = 84

n^{2} – 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.

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