Advertisements
Advertisements
प्रश्न
Prove that if 1 ≤ r ≤ n then `"n" xx ""^(("n" - 1))"C"_("r" - 1) = ""^(("n" - "r" + 1))"C"_("r" - 1)`
उत्तर
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
संबंधित प्रश्न
If nPr = 1680 and nCr = 70, find n and r.
How many triangles can be formed by joining the vertices of a hexagon?
If four dice are rolled, find the number of possible outcomes in which atleast one die shows 2.
The number of diagonals in a polygon of n sides is equal to
The number of 3 letter words that can be formed from the letters of the word ‘NUMBER’ when the repetition is allowed are:
Thirteen guests have participated in a dinner. The number of handshakes that happened in the dinner is:
If nPr = 720 and nCr = 120, find n, r
How many chords can be drawn through 20 points on a circle?
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?
How many ways a committee of six persons from 10 persons can be chosen along with a chair person and a secretary?
Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly three aces in each combination
A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box, if at least one black ball is to be included in the draw?
Find the number of strings of 4 letters that can be formed with the letters of the word EXAMINATION?
How many triangles can be formed by 15 points, in which 7 of them lie on one line and the remaining 8 on another parallel line?
Choose the correct alternative:
The number of ways in which a host lady invite 8 people for a party of 8 out of 12 people of whom two do not want to attend the party together is
Choose the correct alternative:
If 10 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, then the total number of points of intersection are
Choose the correct alternative:
The number of ways of choosing 5 cards out of a deck of 52 cards which include at least one king is
Choose the correct alternative:
If nC4, nC5, nC6 are in AP the value of n can be
