Advertisements
Advertisements
प्रश्न
If P (2n − 1, n) : P (2n + 1, n − 1) = 22 : 7 find n.
Advertisements
उत्तर
P (2n − 1, n):P (2n + 1, n − 1) = 22:7
\[\Rightarrow \frac{\left( 2n - 1 \right)!}{\left( 2n - 1 - n \right)!} \times \frac{\left( 2n + 1 - n + 1 \right)!}{\left( 2n + 1 \right)!} = \frac{22}{7}\]
\[ \Rightarrow \frac{\left( 2n - 1 \right)!}{\left( n - 1 \right)!} \times \frac{\left( n + 2 \right)!}{\left( 2n + 1 \right)!} = \frac{22}{7}\]
\[ \Rightarrow \frac{\left( 2n - 1 \right)!}{\left( n - 1 \right)!} \times \frac{\left( n + 2 \right)\left( n + 1 \right)\left( n \right)\left( n - 1 \right)!}{\left( 2n + 1 \right)\left( 2n \right)\left( 2n - 1 \right)!} = \frac{22}{7}\]
\[ \Rightarrow \frac{\left( n + 2 \right)\left( n + 1 \right)\left( n \right)}{\left( 2n + 1 \right)\left( 2n \right)} = \frac{22}{7}\]
\[ \Rightarrow \frac{\left( n + 2 \right)\left( n + 1 \right)}{2\left( 2n + 1 \right)} = \frac{22}{7}\]
\[ \Rightarrow 7 n^2 + 21n + 14 = 88n + 44\]
\[ \Rightarrow 7 n^2 - 67n - 30 = 0\]
\[ \Rightarrow 7 n^2 - 70n + 3n - 30 = 0\]
\[ \Rightarrow \left( n - 10 \right)\left( 7n + 3 \right) = 0\]
\[ \therefore n = 10 or \frac{- 3}{7}\]
\[\text{Sincencannot be negative, it is equal to10}.\]
APPEARS IN
संबंधित प्रश्न
Convert the following products into factorials:
5 · 6 · 7 · 8 · 9 · 10
Convert the following products into factorials:
(n + 1) (n + 2) (n + 3) ... (2n)
Prove that:
\[\frac{n!}{(n - r)! r!} + \frac{n!}{(n - r + 1)! (r - 1)!} = \frac{(n + 1)!}{r! (n - r + 1)!}\]
If P (5, r) = P (6, r − 1), find r ?
If P (n, 5) = 20. P(n, 3), find n ?
If P (n − 1, 3) : P (n, 4) = 1 : 9, find n.
Prove that:1 . P (1, 1) + 2 . P (2, 2) + 3 . P (3, 3) + ... + n . P (n, n) = P (n + 1, n + 1) − 1.
If n +5Pn +1 =\[\frac{11 (n - 1)}{2}\]n +3Pn, find n.
In how many ways can five children stand in a queue?
How many words, with or without meaning, can be formed by using all the letters of the word 'DELHI', using each letter exactly once?
There are two works each of 3 volumes and two works each of 2 volumes; In how many ways can the 10 books be placed on a shelf so that the volumes of the same work are not separated?
If a denotes the number of permutations of (x + 2) things taken all at a time, b the number of permutations of x things taken 11 at a time and c the number of permutations of x − 11 things taken all at a time such that a = 182 bc, find the value of x.
In how many ways can the letters of the word 'FAILURE' be arranged so that the consonants may occupy only odd positions?
In how many ways can the letters of the word 'STRANGE' be arranged so that
the vowels never come together?
How many words can be formed from the letters of the word 'SUNDAY'? How many of these begin with D?
How many different words can be formed from the letters of the word 'GANESHPURI'? In how many of these words:
the vowels always occupy even places?
How many permutations can be formed by the letters of the word, 'VOWELS', when
each word begins with O and ends with L?
How many permutations can be formed by the letters of the word, 'VOWELS', when
all vowels come together?
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 4 letters are used at a time?
Find the number of words formed by permuting all the letters of the following words:
PAKISTAN
Find the number of words formed by permuting all the letters of the following words:
SERIES
Find the number of words formed by permuting all the letters of the following words:
EXERCISES
In how many ways can the letters of the word 'ALGEBRA' be arranged without changing the relative order of the vowels and consonants?
Find the total number of arrangements of the letters in the expression a3 b2 c4 when written at full length.
How many words can be formed with the letters of the word 'PARALLEL' so that all L's do not come together?
How many words can be formed by arranging the letters of the word 'MUMBAI' so that all M's come together?
How many different signals can be made from 4 red, 2 white and 3 green flags by arranging all of them vertically on a flagstaff?
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 ?
Find the total number of ways in which six ‘+’ and four ‘−’ signs can be arranged in a line such that no two ‘−’ signs occur together.
In how many ways can the letters of the word
"INTERMEDIATE" be arranged so that:the vowels always occupy even places?
Prove that the product of 2n consecutive negative integers is divisible by (2n)!
Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following:
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 number of ways in which 5 red and 4 white balls can be drawn from a bag containing 10 red and 8 white balls.
Write the number of ways in which 12 boys may be divided into three groups of 4 boys each.
