Advertisements
Advertisements
Question
Prove that if 1 ≤ r ≤ n then `"n" xx ""^(("n" - 1))"C"_("r" - 1) = ""^(("n" - "r" + 1))"C"_("r" - 1)`
Solution
To Prove `"n"[""^("n" - 1)"C"_("r" - 1)] = ""^(("n" - "r" + 1))[""^"n""C"_("r" - 1)]`
L.H.S = `"n"[(("n" - 1)!)/(("r" - 1)!("n" - 1 - ("r" - 1))!("n" - 1 - "r" + 1))]`
= `(""("n" - 1)!)/(("r" - 1)!("n" - "r")!) = ("n"!)/(("r" - 1)!("n" - "r")!)` .....(1)
R.H.S = `""^(("n" - "r" + 1))[""^"n""C"_("r" - 1)]`
= `("n" - "r" + 1)[("n"!)/(("r" - 1)!("n" - "r" - 1)!("n" - "r"+ 1))]`
= `("n" - "r" + 1)[("n"!)/(("r" - 1)!("n" -"r" + 1)!)]`
= `(("n" - "r" + 1)"n"!)/(("r" - 1)!("n" - "r" + 1)("n" - "r")!)`
= `("n"!)/(("r" - 1)!("n" - "r")!)` ......(2)
(1) = (2)
⇒ L.H.S = R.H.S
APPEARS IN
RELATED QUESTIONS
How many triangles can be formed by joining the vertices of a hexagon?
Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?
If a polygon has 44 diagonals, find the number of its sides.
A committee of 5 is to be formed out of 6 gents and 4 ladies. In how many ways this can be done when
- atleast two ladies are included.
- atmost two ladies are included.
From 20 raffle tickets in a hat, four tickets are to be selected in order. The holder of the first ticket wins a car, the second a motor cycle, the third a bicycle and the fourth a skateboard. In how many different ways can these prizes be awarded?
There are 10 true or false questions in an examination. Then these questions can be answered in
The value of (5C0 + 5C1) + (5C1 + 5C2) + (5C2 + 5C3) + (5C3 + 5C4) + (5C4 + 5C5) is:
Prove that `""^(2"n")"C"_"n" = (2^"n" xx 1 xx 3 xx ... (2"n" - 1))/("n"!)`
Find the total number of subsets of a set with
[Hint: nC0 + nC1 + nC2 + ... + nCn = 2n] 5 elements
A trust has 25 members. In how many ways can a President, Vice President and a Secretary be selected?
There are 5 teachers and 20 students. Out of them a committee of 2 teachers and 3 students is to be formed. Find the number of ways in which this can be done. Further find in how many of these committees a particular student is excluded?
Find the number of ways of forming a committee of 5 members out of 7 Indians and 5 Americans, so that always Indians will be the majority in the committee
A committee of 7 peoples has to be formed from 8 men and 4 women. In how many ways can this be done when the committee consists of exactly 3 women?
A committee of 7 peoples has to be formed from 8 men and 4 women. In how many ways can this be done when the committee consists of at least 3 women?
There are 11 points in a plane. No three of these lie in the same straight line except 4 points which are collinear. Find the number of triangles that can be formed for which the points are their vertices?
Choose the correct alternative:
The number of rectangles that a chessboard has ______
