Advertisements
Advertisements
Question
If 5 P(4, n) = 6. P (5, n − 1), find n ?
Advertisements
Solution
5 P(4, n) = 6. P (5, n − 1)
5 4Pn = 65Pn
-1\[\Rightarrow 5 \times \frac{4!}{\left( 4 - n \right)!} = 6 \times \frac{5!}{\left( 5 - n + 1 \right)!}\]
\[ \Rightarrow 5 \times \frac{\left( 6 - n \right)!}{\left( 4 - n \right)!} = 6 \times \frac{5!}{4!}\]
\[ \Rightarrow 5 \times \frac{\left( 6 - n \right)\left( 6 - n - 1 \right)\left( 6 - n - 2 \right)!}{\left( 4 - n \right)} = 6 \times \frac{5 \times 4!}{4!}\]
\[ \Rightarrow 5 \times \frac{\left( 6 - n \right)\left( 5 - n \right)\left( 4 - n \right)!}{\left( 4 - n \right)} = 6 \times 5\]
\[ \Rightarrow \left( 6 - n \right)\left( 5 - n \right) = 6\]
\[ \Rightarrow \left( 6 - n \right)\left( 5 - n \right) = 3 \times 2\]
\[\text{On comparing the LHS and the RHS, we get}: \]
\[ \Rightarrow 6 - n = 3\]
\[ \Rightarrow n = 3\]
APPEARS IN
RELATED QUESTIONS
Prove that:
Prove that:
If P (5, r) = P (6, r − 1), find r ?
If P(11, r) = P (12, r − 1) find r.
If P (n, 5) : P (n, 3) = 2 : 1, find n.
Four books, one each in Chemistry, Physics, Biology and Mathematics, are to be arranged in a shelf. In how many ways can this be done?
Find the number of different 4-letter words, with or without meanings, that can be formed from the letters of the word 'NUMBER'.
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 6 items in column A and 6 items in column B. A student is asked to match each item in column A with an item in column B. How many possible, correct or incorrect, answers are there to this question?
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 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?
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?
How many three letter words can be made using the letters of the word 'ORIENTAL'?
Find the number of words formed by permuting all the letters of the following words:
PAKISTAN
How many words can be formed with the letters of the word 'UNIVERSITY', the vowels remaining together?
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 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?
How many different numbers, greater than 50000 can be formed with the digits 0, 1, 1, 5, 9.
How many words can be formed from the letters of the word 'SERIES' which start with S and end with S?
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?
If the letters of the word 'LATE' be permuted and the words so formed be arranged as in a dictionary, find the rank of the word LATE.
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'.
Prove that: 4nC2n : 2nCn = [1 · 3 · 5 ... (4n − 1)] : [1 · 3 · 5 ... (2n − 1)]2.
Evaluate
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:
nCr + 2 · nCr − 1 + nCr − 2 = n + 2Cr.
How many words each of 3 vowels and 2 consonants can be formed from the letters of the word INVOLUTE?
If 35Cn +7 = 35C4n − 2 , then write the values of n.
Write the number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines.
