Advertisements
Advertisements
प्रश्न
Prove that:
Advertisements
उत्तर
\[\text{LHS} = \frac{\left( 2n + 1 \right)!}{n!} \]
\[ = \frac{\left( 2n + 1 \right)\left( 2n \right)\left( 2n - 1 \right) . . . . \left( 4 \right)\left( 3 \right)\left( 2 \right)\left( 1 \right)}{n!}\]
\[ = \frac{\left[ \left( 1 \right)\left( 3 \right)\left( 5 \right) . . . . . . . . . \left( 2n - 1 \right)\left( 2n + 1 \right) \right]\left[ \left( 2 \right)\left( 4 \right)\left( 6 \right) . . . . . . . . . \left( 2n \right) \right]}{n!} \]
\[ = \frac{2^n \left[ \left( 1 \right)\left( 3 \right)\left( 5 \right) . . . . . . . . . \left( 2n - 1 \right)\left( 2n + 1 \right) \right]\left[ \left( 1 \right)\left( 2 \right)\left( 3 \right) . . . . . . . . . \left( n \right) \right]}{n!}\]
\[ = \frac{2^n \left[ \left( 1 \right)\left( 3 \right)\left( 5 \right) . . . . . . . . . \left( 2n - 1 \right)\left( 2n + 1 \right) \right]\left[ n! \right]}{n!}\]
\[ = 2^n \left[ \left( 1 \right)\left( 3 \right)\left( 5 \right) . . . . . . . . . \left( 2n - 1 \right)\left( 2n + 1 \right) \right] = \text{RHS}\]
\[ \text{Hence, proved} . \]
APPEARS IN
संबंधित प्रश्न
Convert the following products into factorials:
3 · 6 · 9 · 12 · 15 · 18
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 (2n − 1, n) : P (2n + 1, n − 1) = 22 : 7 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.
Find the number of different 4-letter words, with or without meanings, that can be formed from the letters of the word 'NUMBER'.
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 6-digit telephone numbers can be constructed with digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 if each number starts with 35 and no digit appears more than once?
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?
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?
In how many ways can the letters of the word 'STRANGE' be arranged so that
the vowels 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 with the letters of word 'SUNDAY'? How many of the words begin with N? How many begin with N and end in Y?
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
there is no restriction on letters?
How many words can be formed out of the letters of the word 'ARTICLE', so that vowels occupy even places?
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.
Find the number of words formed by permuting all the letters of the following words:
PAKISTAN
Find the number of words formed by permuting all the letters of the following words:
EXERCISES
Find the number of words formed by permuting all the letters of the following words:
CONSTANTINOPLE
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 words can be formed from the letters of the word 'SERIES' which start with S and end with S?
How many permutations of the letters of the word 'MADHUBANI' do not begin with M but end with I?
There are three copies each of 4 different books. In how many ways can they be arranged in a shelf?
How many numbers greater than 1000000 can be formed by using the digits 1, 2, 0, 2, 4, 2, 4?
Find the total number of permutations of the letters of the word 'INSTITUTE'.
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.
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:
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, with or without meaning can be formed from the letters of the word 'MONDAY', assuming that no letter is repeated, if (i) 4 letters are used at a time
If 35Cn +7 = 35C4n − 2 , then write the values of n.
Write the maximum number of points of intersection of 8 straight lines in a plane.
