Cryto Eth

How many different ways can the letters of the word butterfly be arranged if the vowels always come together?

In how many different ways can the letters of the word TRAINER be arranged so that the vowels always come together?
A. 1440
B. 120
C. 720
D. 360

Answer

Verified

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 TRAINER, we have to arrange the letters of the word in such a way that all the vowels in the word TRAINER should be together.
The number of vowels in the word TRAINER are = 3 vowels.
The three vowels in the word TRAINER are A, I, and E.
Now these three vowels should always be together and these vowels can be in any order, but they should be together.
Here the three vowels AIE can be arranged in 3 factorial ways, as there are 3 vowels, as given below:
The number of ways the 3 vowels AIE can be arranged is = $3!$
Now arranging the consonants other than the vowels is given by:
As the left out letters in the word TRAINER are TRNR.
The total no. of consonants left out are = 4 consonants.
Now these 4 consonants can be arranged in the following way:
As in the 4 letters TRNR, the letter R is repeated for 2 times, hence the letters TRNR can be arranged in :
$ \Rightarrow \dfrac{{4!}}{{2!}}$
But the letters TRNR are arranged along with the vowels A,I,E, which should be together always but in any order.
Hence we consider the three vowels as a single letter, now TRNR along with AIE can be arranged in:
$ \Rightarrow \dfrac{{5!}}{{2!}}$
But here the vowels can be arranged in $3!$ as already discussed before.
Thus the word TRAINER can be arranged so that the vowels always come together are given below:
$ \Rightarrow \dfrac{{5!}}{{2!}} \times 3! = \dfrac{{120 \times 6}}{2}$
$ \Rightarrow 360$

The number of ways the word TRAINER can be arranged so that the vowels always come together are 360.

Note: Here while solving such kind of problems if there is any word of $n$ letters and a letter is repeating for $r$ times in it, then it can be arranged in $\dfrac{{n!}}{{r!}}$ number of ways. If there are many letters repeating for a distinct number of times, such as a word of $n$ letters and ${r_1}$ repeated items, ${r_2}$ repeated items,…….${r_k}$ repeated items, then it is arranged in $\dfrac{{n!}}{{{r_1}!{r_2}!......{r_k}!}}$ number of ways.

July 30, 2018
4.5K views

IBPS RRB 2018 Prelims : Quants – Day 5

Dear Bankersdaily Aspirants,

IBPS RRB PO & Office Assistant 2018 Preliminary Examination starts from 11th August, 2018. The IBPS RRB Exam 2018 is one of the most expected exam of the year. The reason for the consummate expectation is different from that of the others.

The number of candidates attending the IBPS RRB PO & Office Assistant is high when compared to the other exams like SBI & Insurance Exams. The other important factor is that , no English section is there in the preliminary examination of the IBPS RRB PO & Assistant Exams. This is really a boost to the candidates as many of the aspirants couldn’t get the requisite marks in the English section and this exam is the best bet for the aspirants.

Since only few days are left for the IBPS RRB PO & Assistant Exam 2018 Preliminary Exam , we will be providing daily 20 Questions in Quantitative Aptitude & Reasoning Section to evoke the preparation mode.

Aspirants can answer the Questions by commenting the answers for the Questions in the comments section. We will be providing the answers for the Questions @ 6:00 P.M everyday.

IBPS RRB PO & Assistant 2018 – Prelims Exam – Day 5

Topic : Quantitative Aptitude

Total Number of Questions : 20

Total Time : 15 minutes

D.1-3] Study the following information and answer the given questions:

The table shows the number of two wheelers and four wheelers manufactured and sold 5 different companies.

Company	Number of two wheelers manufactured	Number of two wheelers sold	Number of four wheelers manufactured	Number of four wheelers sold
P	900	560	540	370
Q	1020	600	450	290
R	850	430	420	240
S	980	420	500	330
T	1010	560	660	220

Q.1] Thenumber of four wheeler sold by company P is approximately what percentage of the number of two wheelers sold by company T?

a] 63%

b] 67%

c] 69%

d] 62%

e] 66%

Q.2] What is the difference between the number of two wheelers sold by all companies together and the number of four wheelers unsold by all companies together?

a] 1390

b] 1450

c] 1550

d] 1640

e] 1230

Q.3] If number of two wheelers manufactured by company U is 20% more than the average number of two wheelers unsold by company S and T together, then what is the total number of two wheelers manufactured by company U?

a] 805

b] 1023

c] 1111

d] 606

e] 780

Q.4] A bag contains 4 white balls, 6 pink balls and 8 blue balls, two balls are picked randomly without replacement, what is the probability that one ball is pink colour and another ball is blue colour?

Q.5] th of a number x is 45% of another number y, if the value of y is square of the first two digit even number, what is the difference between x and y?

a] 25