Advertisements
Advertisements
प्रश्न
If \[\frac{(2n)!}{3! (2n - 3)!}\] and \[\frac{n!}{2! (n - 2)!}\] are in the ratio 44 : 3, find n.
Advertisements
उत्तर
\[\frac{(2n)!}{3!(2n - 3)!}: \frac{n!}{2!(n - 2)!} = 44: 3 \]
\[ \Rightarrow \frac{(2n)!}{3!(2n - 3)!} \times \frac{2!(n - 2)!}{n!} = \frac{44}{3}\]
\[ \Rightarrow \frac{(2n)(2n - 1)(2n - 2) [(2n - 3)!]}{3(2!)(2n - 3)!} \times \frac{2!(n - 2)!}{n(n - 1) [(n - 2)!]} = \frac{44}{3}\]
\[ \Rightarrow \frac{(2n)(2n - 1)(2n - 2)}{3} \times \frac{1}{n(n - 1)} = \frac{44}{3}\]
\[ \Rightarrow \frac{(2n)(2n - 1)(2)(n - 1)}{3} \times \frac{1}{n(n - 1)} = \frac{44}{3}\]
\[ \Rightarrow \frac{4(2n - 1)n(n - 1)}{3} \times \frac{1}{n(n - 1)} = \frac{44}{3}\]
\[ \Rightarrow \frac{4(2n - 1)}{3} = \frac{44}{3}\]
\[ \Rightarrow (2n - 1) = 11\]
\[ \Rightarrow 2n = 12\]
\[ \Rightarrow n = 6\]
APPEARS IN
संबंधित प्रश्न
Convert the following products into factorials:
3 · 6 · 9 · 12 · 15 · 18
Prove that: n! (n + 2) = n! + (n + 1)!
If (n + 1)! = 90 [(n − 1)!], find n.
Prove that:
If 5 P(4, n) = 6. P (5, n − 1), find n ?
If P(11, r) = P (12, r − 1) find r.
If P (15, r − 1) : P (16, r − 2) = 3 : 4, find r.
If n +5Pn +1 =\[\frac{11 (n - 1)}{2}\]n +3Pn, find n.
In how many ways can five children stand in a queue?
From among the 36 teachers in a school, one principal and one vice-principal are to be appointed. In how many ways can this be done?
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?
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?
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 'STRANGE' be arranged so that
the vowels come together?
How many different words can be formed from the letters of the word 'GANESHPURI'? In how many of these words:
the letter G always occupies the first place?
How many different words can be formed from the letters of the word 'GANESHPURI'? In how many of these words:
the letters P and I respectively occupy first and last place?
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?
m men and n women are to be seated in a row so that no two women sit together. if m > n then show that the number of ways in which they can be seated as\[\frac{m! (m + 1)!}{(m - n + 1) !}\]
Find the number of words formed by permuting all the letters of the following words:
INDEPENDENCE
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:
CONSTANTINOPLE
How many words can be formed from the letters of the word 'SERIES' which start with S and end with S?
How many permutations of the letters of the word 'MADHUBANI' do not begin with M but end with I?
Find the number of numbers, greater than a million, that can be formed with the digits 2, 3, 0, 3, 4, 2, 3.
How many different arrangements can be made by using all the letters in the word 'MATHEMATICS'. How many of them begin with C? How many of them begin with T?
How many numbers greater than 1000000 can be formed by using the digits 1, 2, 0, 2, 4, 2, 4?
The letters of the word 'SURITI' are written in all possible orders and these words are written out as in a dictionary. Find the rank of the word 'SURITI'.
Prove that the product of 2n consecutive negative integers is divisible by (2n)!
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:
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?
How many words each of 3 vowels and 2 consonants can be formed from the letters of the word INVOLUTE?
Write the number of ways in which 12 boys may be divided into three groups of 4 boys each.
