Advertisements
Advertisements
प्रश्न
Find the number of permutations of n different things taken r at a time such that two specified things occur together?
Advertisements
उत्तर
We have n different things.
We are to select r things at a time such that two specified things occur together.
Remaining things = n - 2
Out of the remaining (n - 2) things, we can select (r - 2) things in n- 2Cr -2ways.
\[ = 2 \left( r - 1 \right)^{n - 2} C_{r - 2} \times \left( r - 2 \right)!\]
\[ = 2 \left( r - 1 \right)^{n - 2} P_{r - 2}\]
APPEARS IN
संबंधित प्रश्न
If (n + 2)! = 60 [(n − 1)!], find n.
Prove that:
\[\frac{n!}{(n - r)! r!} + \frac{n!}{(n - r + 1)! (r - 1)!} = \frac{(n + 1)!}{r! (n - r + 1)!}\]
If P (n, 5) = 20. P(n, 3), find n ?
If P (n − 1, 3) : P (n, 4) = 1 : 9, find n.
If P (n, 5) : P (n, 3) = 2 : 1, find n.
If P (15, r − 1) : P (16, r − 2) = 3 : 4, find r.
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?
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?
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 three-digit numbers are there, with no digit repeated?
How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit 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?
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 letters P and I respectively occupy first and last place?
How many permutations can be formed by the letters of the word, 'VOWELS', when
there is no restriction on letters?
How many permutations can be formed by the letters of the word, 'VOWELS', when
each word begins with O and ends with L?
Find the number of words formed by permuting all the letters of the following words:
EXERCISES
Find the total number of arrangements of the letters in the expression a3 b2 c4 when written at full length.
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.
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 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 relative order of vowels and consonants do not alter?
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:
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.
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.
Find the number of permutations of n distinct things taken r together, in which 3 particular things must occur together.
Write the maximum number of points of intersection of 8 straight lines in a plane.
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.
