Advertisements
Advertisements
Question
Show that (n + 1) `""^"n""P"_"r" = ("n" - "r" + 1) ""^(("n" + 1))"P"_"r"`
Solution
L.H.S. = (n + 1) `""^"n""P"_"r" `
= `("n" + 1) xx ("n!")/(("n" - "r")!)`
= `(("n" + 1)!)/(("n" - "r")!)` .....(I)
= R.H.S. = `("n" - "r" + 1) ""^(("n" + 1))"P"_"r"`
= `("n" - "r" + 1) xx (("n" + 1)!)/(("n" - "r" + 1)!)`
= `(("n" - "r" + 1)("n" + 1)!)/(("n" - "r" + 1)("n" - "r")!)`
= `(("n" + 1)!)/(("n" - "r")!)` .....(II)
From (I) and (II), L.H.S. = R.H.S.
∴ (n + 1)`""^"n""P"_"r" = ("n" - "r" + 1) ""^(("n" + 1))"P"_"r"`
APPEARS IN
RELATED QUESTIONS
Find n, if nP6 : nP3 = 120: 1.
Find m and n if `""^(("m" + "n"))"P"_2` = 56 and `""^(("m" - "n"))"P"_2` = 12
How many 4 letter words can be formed using letters in the word MADHURI if letters cannot be repeated?
Find the number of arranging 11 distinct objects taken 4 at a time so that a specified object always occurs.
Find the number of arranging 11 distinct objects taken 4 at a time so that a specified object never occurs.
Find the number of permutations of letters of the following word: SHANTARAM
Find the number of permutations of letters of the following word: REPRESENT
Find the number of permutations of letters of the following word: COMBINE
How many distinct 5 digit numbers can be formed using the digits 3, 2, 3, 2, 4, 5
Find the number of distinct numbers formed using the digits 3, 4, 5, 6, 7, 8, 9, so that odd positions are occupied by odd digits.
Find m and n, if (m+n)P2 = 56 and (m-n)P2 = 12
Find r, if 12Pr–2 : 11Pr–1 = 3 : 14
Show that (n + 1) (nPr) = (n – r + 1) [(n+1)Pr]
A code word is formed by two different English letters followed by two non-zero distinct digits. Find the number of such code words. Also, find the number of such code words that end with an even digit.
Find the number of arranging 11 distinct objects taken 4 at a time so that a specified object always occurs
Find the number of arranging 11 distinct objects taken 4 at a time so that a specified object never occurs
