Advertisements
Advertisements
Question
If P(11, r) = P (12, r − 1) find r.
Advertisements
Solution
P(11, r) = P (12, r − 1)
\[ \Rightarrow \frac{\left( 13 - r \right)}{\left( 11 - r \right)!} = \frac{12!}{11!}\]
\[ \Rightarrow \frac{\left( 13 - r \right)\left( 12 - r \right)\left( 11 - r \right)!}{\left( 11 - r \right)!} = \frac{12 \times 11!}{11!}\]
\[ \Rightarrow \left( 13 - r \right)\left( 12 - r \right) = 12\]
\[ \Rightarrow \left( 13 - r \right)\left( 12 - r \right) = 4 \times 3\]
\[\text{On comparing the two sides, we get}: \]
\[13 - r = 4\]
\[ \Rightarrow r = 9\]
APPEARS IN
RELATED QUESTIONS
Convert the following products into factorials:
1 · 3 · 5 · 7 · 9 ... (2n − 1)
Prove that: n! (n + 2) = n! + (n + 1)!
If (n + 2)! = 60 [(n − 1)!], find n.
If (n + 1)! = 90 [(n − 1)!], find n.
If (n + 3)! = 56 [(n + 1)!], find n.
Prove that:
If P (9, r) = 3024, find r.
If P (n, 4) = 12 . P (n, 2), find n.
If P (n, 5) : P (n, 3) = 2 : 1, find n.
If P (15, r − 1) : P (16, r − 2) = 3 : 4, find r.
In how many ways can five children stand in a queue?
In how many ways can 6 boys and 5 girls be arranged for a group photograph if the girls are to sit on chairs in a row and the boys are to stand in a row behind them?
How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated?
Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5, if no digit is repeated? How many of these will be even?
How many words can be formed out of the letters of the word 'ARTICLE', so that vowels occupy even places?
How many words (with or without dictionary meaning) can be made from the letters in the word MONDAY, assuming that no letter is repeated, if all letters are used at a time.
Find the number of words formed by permuting all the letters of the following words:
PAKISTAN
In how many ways can the letters of the word 'ALGEBRA' be arranged without changing the relative order of the vowels and consonants?
How many words can be formed with the letters of the word 'UNIVERSITY', the vowels remaining together?
How many numbers can be formed with the digits 1, 2, 3, 4, 3, 2, 1 so that the odd digits always occupy the odd places?
In how many ways can the letters of the word 'ARRANGE' be arranged so that the two R's are never together?
In how many ways can the letters of the word ASSASSINATION be arranged so that all the S's are together?
Find the total number of permutations of the letters of the word 'INSTITUTE'.
If the permutations of a, b, c, d, e taken all together be written down in alphabetical order as in dictionary and numbered, find the rank of the permutation debac ?
Prove that the product of 2n consecutive negative integers is divisible by (2n)!
For all positive integers n, show that 2nCn + 2nCn − 1 = `1/2` 2n + 2Cn+1
Prove that: 4nC2n : 2nCn = [1 · 3 · 5 ... (4n − 1)] : [1 · 3 · 5 ... (2n − 1)]2.
Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following:
Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following:
n · n − 1Cr − 1 = (n − r + 1) nCr − 1
How many words, with or without meaning can be formed from the letters of the word 'MONDAY', assuming that no letter is repeated, if all letters are used but first letter is a vowel?
Find the number of permutations of n distinct things taken r together, in which 3 particular things must occur together.
Find the number of permutations of n different things taken r at a time such that two specified things occur together?
Write the value of\[\sum^6_{r = 1} \ ^{56 - r}{}{C}_3 + \ ^ {50}{}{C}_4\]
Write the number of ways in which 12 boys may be divided into three groups of 4 boys each.
