Advertisements
Advertisements
प्रश्न
If n +5Pn +1 =\[\frac{11 (n - 1)}{2}\]n +3Pn, find n.
Advertisements
उत्तर
n +5Pn +1 =11(n-1)2n +3Pn
\[ \Rightarrow \frac{\left( n + 5 \right)!}{4!} = \frac{11\left( n - 1 \right)}{2} \times \frac{\left( n + 3 \right)!}{3!}\]
\[ \Rightarrow \frac{\left( n + 5 \right)!}{\left( n + 3 \right)!} = \frac{11\left( n - 1 \right)}{2} \times \frac{4!}{3!}\]
\[ \Rightarrow \frac{\left( n + 5 \right)\left( n + 4 \right)\left( n + 3 \right)!}{\left( n + 3 \right)!} = \frac{11\left( n - 1 \right)}{2} \times \frac{4 \times 3!}{3!}\]
\[ \Rightarrow \left( n + 5 \right)\left( n + 4 \right) = 22\left( n - 1 \right)\]
\[ \Rightarrow n^2 + 9n + 20 = 22n - 22\]
\[ \Rightarrow n^2 - 13n + 42 = 0\]
\[ \Rightarrow n = 7, 6\]
APPEARS IN
संबंधित प्रश्न
Convert the following products into factorials:
3 · 6 · 9 · 12 · 15 · 18
Convert the following products into factorials:
(n + 1) (n + 2) (n + 3) ... (2n)
Convert the following products into factorials:
1 · 3 · 5 · 7 · 9 ... (2n − 1)
If P (9, r) = 3024, 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 P (15, r − 1) : P (16, r − 2) = 3 : 4, find r.
In how many ways can five children stand in a queue?
Four letters E, K, S and V, one in each, were purchased from a plastic warehouse. How many ordered pairs of letters, to be used as initials, can be formed from them?
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?
Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5, if no digit is repeated? How many of these will be even?
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 from the letters of the word 'GANESHPURI'? In how many of these words:
the letter G always occupies the first place?
How many permutations can be formed by the letters of the word, 'VOWELS', when
each word begins with E?
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?
Find the number of words formed by permuting all the letters of the following words:
INDEPENDENCE
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?
How many words can be formed by arranging the letters of the word 'MUMBAI' so that all M's 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 different signals can be made from 4 red, 2 white and 3 green flags by arranging all of them vertically on a flagstaff?
A biologist studying the genetic code is interested to know the number of possible arrangements of 12 molecules in a chain. The chain contains 4 different molecules represented by the initials A (for Adenine), C (for Cytosine), G (for Guanine) and T (for Thymine) and 3 molecules of each kind. How many different such arrangements are possible?
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.
In how many ways can the letters of the word
"INTERMEDIATE" be arranged so that:the vowels always occupy even places?
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'?
Prove that the product of 2n consecutive negative integers is divisible by (2n)!
Prove that: 4nC2n : 2nCn = [1 · 3 · 5 ... (4n − 1)] : [1 · 3 · 5 ... (2n − 1)]2.
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
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?
Write the value of\[\sum^6_{r = 1} \ ^{56 - r}{}{C}_3 + \ ^ {50}{}{C}_4\]
Write the number of ways in which 12 boys may be divided into three groups of 4 boys each.
