Advertisements
Advertisements
प्रश्न
If P (15, r − 1) : P (16, r − 2) = 3 : 4, find r.
Advertisements
उत्तर
P (15, r − 1):P (16, r − 2) = 3:4
\[\Rightarrow \frac{15!}{\left( 15 - r + 1 \right)!} \times \frac{(16 - r + 2)!}{16!} = \frac{3}{4}\]
\[ \Rightarrow \frac{15!}{\left( 16 - r \right)!} \times \frac{\left( 18 - r \right)!}{16 \times 15!} = \frac{3}{4}\]
\[ \Rightarrow \frac{\left( 18 - r \right)\left( 17 - r \right)\left( 16 - r \right)!}{\left( 16 - r \right)!\left( 16 \right)} = \frac{3}{4}\]
\[ \Rightarrow \left( 18 - r \right)\left( 17 - r \right) = 12\]
\[ \Rightarrow \left( 18 - r \right)\left( 17 - r \right) = 4 \times 3\]
\[\text{On comparing the LHS and the RHS in above expression, we get}: \]
\[ \Rightarrow 18 - r = 14\]
\[ \Rightarrow r = 14\]
APPEARS IN
संबंधित प्रश्न
If (n + 2)! = 60 [(n − 1)!], find n.
Prove that:
Prove that:
\[\frac{n!}{(n - r)! r!} + \frac{n!}{(n - r + 1)! (r - 1)!} = \frac{(n + 1)!}{r! (n - r + 1)!}\]
If nP4 = 360, find the value of n.
If P(11, r) = P (12, r − 1) find r.
If P (n, 5) : P (n, 3) = 2 : 1, 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.
How many three-digit numbers are there, with distinct digits, with each digit odd?
How many words, with or without meaning, can be formed by using the letters of the word 'TRIANGLE'?
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 'STRANGE' be arranged so that
the vowels come together?
In how many ways can the letters of the word 'STRANGE' be arranged so that
the vowels never come together?
In how many ways can the letters of the word 'STRANGE' be arranged so that
the vowels occupy only the odd places?
How many words can be formed out of the letters of the word, 'ORIENTAL', so that the vowels always occupy the odd places?
How many different words can be formed with the letters of word 'SUNDAY'? How many of the words begin with N? How many begin with N and end in Y?
In how many ways can a lawn tennis mixed double be made up from seven married couples if no husband and wife play in the same set?
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 with the letters of the word 'UNIVERSITY', the vowels remaining together?
How many words can be formed with the letters of the word 'PARALLEL' so that all L's do not come 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?
How many number of four digits can be formed with the digits 1, 3, 3, 0?
Find the number of numbers, greater than a million, that can be formed with the digits 2, 3, 0, 3, 4, 2, 3.
If the letters of the word 'MOTHER' are written in all possible orders and these words are written out as in a dictionary, find the rank of the word 'MOTHER'.
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 ?
The letters of the word 'ZENITH' are written in all possible orders. How many words are possible if all these words are written out as in a dictionary? What is the rank of the word 'ZENITH'?
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
Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following:
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 at a time
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.
