Question

Number of four letter words that can be formed with the letters of the word ‘PHYSICS’ which contain Y
A. 288
B. 480
C. 240
D. None of these.

Answer 1

Hint: To solve this problem we have to know about the concept of permutations and combinations. But here a simple concept is used. In any given word, the number of ways we can arrange the word by jumbling the letters is the number of letters present in the word factorial. Here factorial of any number is the product of that number and all the numbers less than that number till 1.
$ \Rightarrow n! = n(n - 1)(n - 2).......1$

Complete step by step answer:
Given the word PHYSICS, we are asked to form a four letter word using the letters from the word ‘PHYSICS’ which contains Y in it.
Here using the concept of permutations and combinations.
Permutations formula used here is given by:
When given n observations and from those we have to arrange r observations, is given by:
$ \Rightarrow {}^n{P_r} = \dfrac{{n!}}{{(n - r)!}}$
Here given the word ‘PHYSICS’, which has 7 letters in it, and we have to form a four letter word using the letters in it, which should also contain the letter Y.
We know that if we have to form a four letter then it is just arranging 4 letters from a nine letter word, which can be mathematically expressed as:
$ \Rightarrow {}^9{P_4} = \dfrac{{9!}}{{(9 - 4)!}}$
$ \Rightarrow {}^9{P_4} = \dfrac{{9!}}{{5!}}$ ways.
But here we have to include Y in it, hence we have to arrange only 3 letters from the remaining 6 letters in the word ‘PHYSICS’.
This letter Y can be arranged in 4 ways, as there are 4 blanks to form a four letter word.
_ _ _ Y
Hence there are only three spots left to be arranged from the remaining 6 letter word, as the remaining word is ‘PHSICS’, as Y is already included.
Now the 6 letters can be arranged in 3 spots as given:
$ \Rightarrow {}^6{P_3}$
Now combining of what we have found out till is that the must be included Y letter can be arranged in 4 ways, and from the remaining 6 letters, arranging them in the three spots of the four letter word is given by:
$ \Rightarrow {}^6{P_3} \times 4$
$ \Rightarrow \dfrac{{6!}}{{(6 - 3)!}} \times 4$
$ \Rightarrow \dfrac{{6!}}{{3!}} \times 4$
On expanding the above expression, is given by:
$ \Rightarrow \dfrac{{6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{3 \times 2 \times 1}} \times 4$
$ \Rightarrow 6 \times 5 \times 4 \times 4$
$ \Rightarrow 30 \times 16$
$ \Rightarrow 480$

The no. of four letter words that can be formed with the letters of the word ‘PHYSICS’ which contain Y are 480.

Note: While solving this problem please note that the arrangements made after involving the letter Y are selecting three letters from the remaining 6 letters of the word PHYSICS because the letter Y is already involved in the about to form four letter word. Hence arranging and picking three letters from the six letter word ‘PHYSICS’ is given by ${}^6{P_3}$.