Advertisements
Advertisements
Question
Find r if `""^5P_r = ""^6P_(r-1)`
Advertisements
Solution
`""^5P_r = ""^6P_(r-1)`
⇒ `(5!)/((5 - r)!) = (6!)/((6 - r + 1)!)`
⇒ `(5!)/((5 - r)!) = (6 xx 5!)/((7 - r )!)`
⇒ `1/((5 - r)!) = 6/((7 - r)(6 - r)(5 - r)!)`
⇒ 1 = `6/((7 - r)(6 - r))`
⇒ (7 - r)(6 - r) = 6
⇒ 42 - 7r - 6r + r2 - 6 = 0
⇒ r2 - 13r + 36 = 0
⇒ r2 - 4r - 9r + 36 = 0
⇒ r(r - 4 ) -9 (r - 4) = 0
⇒ r(r - 4)(r - 9) = 0
⇒ (r - 4) = 0 or (r - 9) = 0
⇒ r = 4 or r = 9
It is known that, `""^nP_r = (n!)/((n - r)!) 0 ≤ r ≤ n`
∴ 0 ≤ r ≤ 5
Hence, r ≠ 9
∴ r = 4
APPEARS IN
RELATED QUESTIONS
Is 3! + 4! = 7!?
if `1/(6!) + 1/(7!) = x/(8!)`, find x
Evaluate `(n!)/((n-r)!)` when n = 6, r = 2
Find r if `""^5P_r = 2^6 P_(r-1)`
In how many ways can the letters of the word PERMUTATIONS be arranged if the vowels are all together.
Find x in each of the following:
Which of the following are true:
(2 +3)! = 2! + 3!
In how many ways can three jobs I, II and III be assigned to three persons A, B and C if one person is assigned only one job and all are capable of doing each job?
A coin is tossed three times and the outcomes are recorded. How many possible outcomes are there? How many possible outcomes if the coin is tossed four times? Five times? n times?
Three dice are rolled. Find the number of possible outcomes in which at least one die shows 5 ?
In how many ways can 5 different balls be distributed among three boxes?
In how many ways can 7 letters be posted in 4 letter boxes?
In how many ways can 4 letters be posted in 5 letter boxes?
Write the number of 5 digit numbers that can be formed using digits 0, 1 and 2 ?
In how many ways 4 women draw water from 4 taps, if no tap remains unused?
Write the number of arrangements of the letters of the word BANANA in which two N's come together.
Write the number of ways in which 6 men and 5 women can dine at a round table if no two women sit together ?
The number of arrangements of the letters of the word BHARAT taking 3 at a time is
English alphabet has 11 symmetric letters that appear same when looked at in a mirror. These letters are A, H, I, M, O, T, U, V, W, X and Y. How many symmetric three letters passwords can be formed using these letters?
Evaluate `("n"!)/("r"!("n" - "r")!)` when n = 5 and r = 2.
If (n+2)! = 60[(n–1)!], find n
How many numbers lesser than 1000 can be formed using the digits 5, 6, 7, 8, and 9 if no digit is repeated?
Find the rank of the word ‘CHAT’ in the dictionary.
Evaluate the following.
`(3! + 1!)/(2^2!)`
The possible outcomes when a coin is tossed five times:
The greatest positive integer which divide n(n + 1) (n + 2) (n + 3) for all n ∈ N is:
Suppose 8 people enter an event in a swimming meet. In how many ways could the gold, silver and bronze prizes be awarded?
How many ways can the product a2 b3 c4 be expressed without exponents?
In how many ways can the letters of the word SUCCESS be arranged so that all Ss are together?
A coin is tossed 8 times, how many different sequences of heads and tails are possible?
How many strings are there using the letters of the word INTERMEDIATE, if vowels are never together
Choose the correct alternative:
The product of r consecutive positive integers is divisible b
Choose the correct alternative:
If Pr stands for rPr then the sum of the series 1 + P1 + 2P2 + 3P3 + · · · + nPn is
In how many ways can 5 children be arranged in a line such that two particular children of them are never together.
In how many ways 3 mathematics books, 4 history books, 3 chemistry books and 2 biology books can be arranged on a shelf so that all books of the same subjects are together.
The number of signals that can be sent by 6 flags of different colours taking one or more at a time is ______.
Let b1, b2, b3, b4 be a 4-element permutation with bi ∈ {1, 2, 3, .......,100} for 1 ≤ i ≤ 4 and bi ≠ bj for i ≠ j, such that either b1, b2, b3 are consecutive integers or b2, b3, b4 are consecutive integers. Then the number of such permutations b1, b2, b3, b4 is equal to ______.
Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Determine the number of words which have at least one letter repeated.
