Advertisements
Advertisements
प्रश्न
Convert the following products into factorials:
1 · 3 · 5 · 7 · 9 ... (2n − 1)
उत्तर
\[\ \left( 1 \right)\left( 3 \right)\left( 5 \right) . . . . . . . . . \left( 2n - 1 \right) = \frac{\left[ \left( 1 \right)\left( 3 \right)\left( 5 \right) . . . . . . . . . \left( 2n - 1 \right) \right]\left[ \left( 2 \right)\left( 4 \right)\left( 6 \right) . . . . . . . . . \left( 2n \right) \right]}{\left[ \left( 2 \right)\left( 4 \right)\left( 6 \right) . . . . . . . . . \left( 2n \right) \right]} \]
\[ = \frac{\left[ \left( 1 \right)\left( 2 \right)\left( 3 \right)\left( 4 \right)\left( 5 \right) . . . . . . . . . \left( 2n - 1 \right)\left( 2n \right) \right]}{2^n \left[ \left( 1 \right)\left( 2 \right)\left( 3 \right) . . . . . . . . . \left( n \right) \right]}\]
\[ = \frac{\left( 2n \right)!}{2^n n!}\]
APPEARS IN
संबंधित प्रश्न
Convert the following products into factorials:
3 · 6 · 9 · 12 · 15 · 18
Prove that: n! (n + 2) = n! + (n + 1)!
If (n + 2)! = 60 [(n − 1)!], find n.
If (n + 1)! = 90 [(n − 1)!], find n.
If P (n, 5) = 20. P(n, 3), find n ?
If P(11, r) = P (12, r − 1) find r.
If P (2n − 1, n) : P (2n + 1, n − 1) = 22 : 7 find n.
If P (n, 5) : P (n, 3) = 2 : 1, find n.
If n +5Pn +1 =\[\frac{11 (n - 1)}{2}\]n +3Pn, find n.
In how many ways can five children stand in a queue?
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?
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?
In how many ways can the letters of the word 'STRANGE' be arranged so that
the vowels come together?
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 permutations can be formed by the letters of the word, 'VOWELS', when
each word begins with O and ends with L?
How many words can be formed out of the letters of the word 'ARTICLE', so that vowels occupy even places?
In how many ways can a lawn tennis mixed double be made up from seven married couples if no husband and wife play in the same set?
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 4 letters are used at a time?
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 but first is vowel.
In how many ways can the letters of the word 'ALGEBRA' be arranged without changing the relative order of the vowels and consonants?
Find the total number of arrangements of the letters in the expression a3 b2 c4 when written at full length.
How many words can be formed by arranging the letters of the word 'MUMBAI' so that all M's come together?
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 numbers, greater than 50000 can be formed with the digits 0, 1, 1, 5, 9.
How many words can be formed from the letters of the word 'SERIES' which start with S and end with S?
Find the number of numbers, greater than a million, that can be formed with the digits 2, 3, 0, 3, 4, 2, 3.
In how many ways can the letters of the word ASSASSINATION be arranged so that all the S's are together?
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'.
In how many ways can the letters of the word
"INTERMEDIATE" be arranged so that:the vowels always occupy even places?
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
How many words, with or without meaning can be formed from the letters of the word 'MONDAY', assuming that no letter is repeated, if all letters are used but first letter is a vowel?
If 35Cn +7 = 35C4n − 2 , then write the values of n.
