Question

The number of arrangements that can be formed with the letters of the word ORDINATE, so that the vowels occupy odd places is ${k^2}$ then the greatest prime which can divide k is _____

Answer 1

Hint: The word ORDINATE has 8 letters in total. In which there are 4 vowels and 4 consonants. So the odd places in the word are 1st, 3rd, 5th and 7th places. These 4 places must be occupied with the 4 vowels and the remaining 4 places must be occupied with the 4 consonants. Let the no. of vowels be x and no. of vowels be y, then the no. of arrangements that can be formed with the word ORDINATE with vowels at odd places will be ${}^4{P_x} \times {}^4{P_y}$. Equate this number with ${k^2}$ and find the value of k. Prime factorize k to find its greatest prime factor (divisor).

Complete step-by-step answer:
We are given to find the no. of arrangements that can be formed with the word ORDINATE with vowels at odd places.
The no. of vowels in ORDINATE is 4; O, I, A and E.
The no. of consonants in ORDINATE is 4; R, D, N and T.
The odd places for the 8 letters in ORDINATE are 1st, 3rd, 5th and 7th.
The 4 vowels should occupy these 4 odd places and other 4 places must be occupied by the 4 consonants. Use permutations because we have to arrange the letters in an order such that the vowels occupy odd places.
The no. of ways that can be formed are ${}^4{P_4} \times {}^4{P_4}$
$
  {}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}} \\
   \to {}^4{P_4} = \dfrac{{4!}}{{\left( {4 - 4} \right)!}} = \dfrac{{4!}}{{0!}} \\
  0! = 1 \\
   \to {}^4{P_4} = \dfrac{{4!}}{1} = 4 \times 3 \times 2 \times 1 = 24 \\
  No.ofways = {}^4{P_4} \times {}^4{P_4} = 24 \times 24 = 576 \\
$
This number 576 is equal to ${k^2}$ as given in the question.
$
   \to {k^2} = 576 = {24^2} \\
  \therefore k = 24 \\
$
The value of k is 24.
Now we are Prime factorizing k (24).
$24 = 2 \times 2 \times 2 \times 3$
Therefore, the greatest prime factor which can divide 24 is 3.

Note: A Permutation is arranging the objects in order. Combinations are the way of selecting the objects from a group of objects or collection. When the order of the objects does not matter then it should be considered as Combination and when the order matters then it should be considered as Permutation. Do not confuse using a permutation, when required, instead of a combination and vice-versa.