Advertisements
Advertisements
प्रश्न
Find r if `""^5P_r = 2^6 P_(r-1)`
Advertisements
उत्तर
`""^5P_r = 2^6 P_(r-1)`
⇒ `(5!)/((5 - r)!) = 2 xx (6!)/((6 - r + 1)!)`
⇒ `(5!)/((5 - r)!) = (2 xx 6!)/((7 - r)!)`
⇒ `(5!)/((5 - r)!) = (2 xx 6 xx 5!)/((7 - r)(6 - r)(5 - r)!)`
⇒ 1 = `(2 xx 6)/((7 - r)(6 - r))`
⇒ (7 - r)(6 - r) = 12
⇒ 42 - 6r - 7r + r2 = 12
⇒ r2 - 13r + 30 = 0
⇒ r2 - 3r - 10r + 30 = 0
⇒ r(r - 3) - 10(r - 3) = 0
⇒ (r - 3)(r - 10) = 0
⇒ (r - 3) = 0 or (r - 10) = 0
⇒ r = 3 or r = 10
It is known that `""^nP_r = (n!)/((n - r)!) 0 ≤ r ≤ n`
∴0 ≤ r ≤ 5
Hence, r ≠ 10
∴ r = 3
APPEARS IN
संबंधित प्रश्न
Evaluate 4! – 3!
How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated?
How many 4-digit numbers are there with no digit repeated?
Find r if `""^5P_r = 2^6 P_(r-1)`
Find r if `""^5P_r = ""^6P_(r-1)`
Which of the following are true:
(2 +3)! = 2! + 3!
Which of the following are true:
(2 × 3)! = 2! × 3!
A coin is tossed three times and the outcomes are recorded. How many possible outcomes are there? How many possible outcomes if the coin is tossed four times? Five times? n times?
In how many ways can 5 different balls be distributed among three boxes?
There are 10 lamps in a hall. Each one of them can be switched on independently. Find the number of ways in which the hall can be illuminated ?
Evaluate each of the following:
8P3
Evaluate each of the following:
6P6
Write the number of ways in which 7 men and 7 women can sit on a round table such that no two women sit together ?
Write the number of words that can be formed out of the letters of the word 'COMMITTEE' ?
The number of permutations of n different things taking r at a time when 3 particular things are to be included is
If the letters of the word KRISNA are arranged in all possible ways and these words are written out as in a dictionary, then the rank of the word KRISNA is
The number of arrangements of the letters of the word BHARAT taking 3 at a time is
The number of ways in which the letters of the word ARTICLE can be arranged so that even places are always occupied by consonants is
How many 6-digit telephone numbers can be constructed with the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 if each numbers starts with 35 and no digit appear more than once?
- In how many ways can 8 identical beads be strung on a necklace?
- In how many ways can 8 boys form a ring?
If `""^(("n" – 1))"P"_3 : ""^"n""P"_4` = 1 : 10 find n
Three men have 4 coats, 5 waist coats and 6 caps. In how many ways can they wear them?
A test consists of 10 multiple choice questions. In how many ways can the test be answered if each question has four choices?
A student appears in an objective test which contain 5 multiple choice questions. Each question has four choices out of which one correct answer.
How will the answer change if each question may have more than one correct answers?
8 women and 6 men are standing in a line. In how many arrangements will all 6 men be standing next to one another?
Each of the digits 1, 1, 2, 3, 3 and 4 is written on a separate card. The six cards are then laid out in a row to form a 6-digit number. How many of these 6-digit numbers are divisible by 4?
If the letters of the word GARDEN are permuted in all possible ways and the strings thus formed are arranged in the dictionary order, then find the ranks of the words
DANGER
Find the sum of all 4-digit numbers that can be formed using digits 1, 2, 3, 4, and 5 repetitions not allowed?
Choose the correct alternative:
If Pr stands for rPr then the sum of the series 1 + P1 + 2P2 + 3P3 + · · · + nPn is
The number of arrangements of the letters of the word BANANA in which two N's do not appear adjacently is ______.
If all permutations of the letters of the word AGAIN are arranged in the order as in a dictionary. What is the 49th word?
Suppose m men and n women are to be seated in a row so that no two women sit together. If m > n, show that the number of ways in which they can be seated is `(m!(m + 1)!)/((m - n + 1)1)`
Ten different letters of alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have atleast one letter repeated is ______.
Find the number of different words that can be formed from the letters of the word ‘TRIANGLE’ so that no vowels are together
A five-digit number divisible by 3 is to be formed using the numbers 0, 1, 2, 3, 4 and 5 without repetitions. The total number of ways this can be done is ______.
The number of words which can be formed out of the letters of the word ARTICLE, so that vowels occupy the even place is ______.
The number of permutations of n different objects, taken r at a line, when repetitions are allowed, is ______.
How many words (with or without dictionary meaning) can be made from the letters of the word MONDAY, assuming that no letter is repeated, if
| C1 | C2 |
| (a) 4 letters are used at a time | (i) 720 |
| (b) All letters are used at a time | (ii) 240 |
| (c) All letters are used but the first is a vowel | (iii) 360 |
Number of words from the letters of the words BHARAT in which B and H will never come together is ______.
