English

If 5 P(4, N) = 6. P (5, N − 1), Find N ?

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)
4Pn = 65Pn

-1\[\Rightarrow 5 \times \frac{4!}{\left( 4 - n \right)!} = 6 \times \frac{5!}{\left( 5 - n + 1 \right)!}\]
\[ \Rightarrow 5 \times \frac{\left( 6 - n \right)!}{\left( 4 - n \right)!} = 6 \times \frac{5!}{4!}\]
\[ \Rightarrow 5 \times \frac{\left( 6 - n \right)\left( 6 - n - 1 \right)\left( 6 - n - 2 \right)!}{\left( 4 - n \right)} = 6 \times \frac{5 \times 4!}{4!}\]
\[ \Rightarrow 5 \times \frac{\left( 6 - n \right)\left( 5 - n \right)\left( 4 - n \right)!}{\left( 4 - n \right)} = 6 \times 5\]
\[ \Rightarrow \left( 6 - n \right)\left( 5 - n \right) = 6\]
\[ \Rightarrow \left( 6 - n \right)\left( 5 - n \right) = 3 \times 2\]
\[\text{On comparing the LHS and the RHS, we get}: \]
\[ \Rightarrow 6 - n = 3\]
\[ \Rightarrow n = 3\]

shaalaa.com
Factorial N (N!) Permutations and Combinations
  Is there an error in this question or solution?
Chapter 16: Permutations - Exercise 16.3 [Page 28]

APPEARS IN

R.D. Sharma Mathematics [English] Class 11
Chapter 16 Permutations
Exercise 16.3 | Q 3 | Page 28

RELATED QUESTIONS

Prove that: n! (n + 2) = n! + (n + 1)!


Prove that: 

\[\frac{n!}{(n - r)!}\] = n (n − 1) (n − 2) ... (n − (r − 1))

If P (5, r) = P (6, r − 1), find r ?


If P(11, r) = P (12, r − 1) find r.


If P (n, 4) = 12 . P (n, 2), find n.


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 (nn) = P (n + 1, n + 1) − 1.


If P (15, r − 1) : P (16, r − 2) = 3 : 4, find r.


In how many ways can five children stand in a queue?


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 words, with or without meaning, can be formed by using all the letters of the word 'DELHI', using each letter exactly once?


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.


How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated?


How many 3-digit even number can be made using the digits 1, 2, 3, 4, 5, 6, 7, if no digits is repeated?


All the letters of the word 'EAMCOT' are arranged in different possible ways. Find the number of arrangements in which no two vowels are adjacent to each other.


In how many ways can the letters of the word 'FAILURE' be arranged so that the consonants may occupy only odd positions?


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 vowels are always 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 all 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:
INTERMEDIATE


Find the number of words formed by permuting all the letters of the following words:

RUSSIA


Find the number of words formed by permuting all the letters of the following words:
SERIES


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?


In how many ways can the letters of the word 'ARRANGE' be arranged so that the two R's are never together?


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?


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'.


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'.


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.


Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following:

 nCr + 2 · nCr − 1 + nCr − 2 = n + 2Cr.


There are 10 persons named\[P_1 , P_2 , P_3 , . . . . , P_{10}\]
Out of 10 persons, 5 persons are to be arranged in a line such that in each arrangement P1 must occur whereas P4 and P5 do not occur. Find the number of such possible arrangements.


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.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×