How many ways are there to arrange the letter in the word GARDEN with the wall in alphabetical order?

(1) 120

(2) 240

(3) 360

(4) 480

Answer: (3) 360

Solution:

Given the words, GARDEN

Total number of letters = 6

Total number of ways in which all 6 letters can be arranged = 6! Ways

Total number of ways in which two vowels (A, E) can be arranged between them = 2! Ways

Therefore,

Total number of ways = 6!/2! = 360

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How many 3-digit odd numbers can be formed from the digits 1,2,3,4,5,6 if:
(a) the digits can be repeated (b) the digits cannot be repeated?

(a) Number of digits available = 6

Number of places [(x), (y) and (z)] for them = 3Repetition is allowed and the 3-digit numbers formed are oddNumber of ways in which box (x) can be filled = 3 (by 1, 3 or 5 as the numbers formed are to be odd)

m = 3

Number of ways of filling box (y) = 6 (∴ Repetition is allowed)

n = 6

Number of ways of filling box (z) = 6 (∵ Repetition is allowed)

p = 6

∴ Total number of 3-digit odd numbers formed = m x n x p = 3 x 6 x 6 = 108(b) Number of ways of filling box (x) = 3 (only odd numbers are to be in this box )

m = 3

Number of ways of filling box (y) = 5 (∵ Repetition is not allowed)

n = 5

Number of ways of filling box (z) = 4 (∵ Repetition is not allowed)

p = 4

∴ Total number of 3-digit odd numbers formed

= m x n x p = 3 x 5 x 4 = 60.