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The correct ANSWER is (a) 1

The best explanation: ^4PR = 4*^5Pr-1

=> \(\frac{4!}{(4-r)!} = 4*\frac{5!}{(5-r+1)!}\)

=> \(\frac{(6-r)!}{(4-r)!} = 4*\frac{5!}{4!}\)

=> (6-r) (5-r) = 4*5

=> r=1.

The correct ANSWER is (c) 120

Easy explanation: IMAGE is a 5 letters word. We have to arrange all 5 letters of the word IMAGE without REPETITION. So, total PERMUTATIONS are ^nPr = ^5P5 = 5! = 5.4.3.2.1 = 120.

Right answer is (d) N^R

The best explanation: The NUMBER of permutations of n different objects TAKEN r at a time, where repetition is allowed is n*n*n*n*n……………. r TIMES = n^r.

The correct choice is (b) 6

To explain I would say: By the fundamental principle of COUNTING, if an event can occur in ‘m’ different ways, following which another event can occur in ‘n’ different ways, then the total numbers of occurrence of the EVENTS in the given order is m*n.

So, if pencil can be taken in 2 ways and ERASER can be taken in 3 ways then number of ways in which he can TAKE a pencil and eraser are 2*3 = 6.

Right choice is (c) 30

Best explanation: Out of 5 males, 2 males can be selected in ^5C2 ways.

^5C2 = \(\frac{5!}{(5-2)! 2!} = \frac{5*4*3!}{(3)! 2!} = \frac{20}{2}\) = 10 ways.

Out of 3 females, 2 females can be selected in ^3C2 ways .

^3C2 = \(\frac{3!}{(3-2)! 2!} = \frac{3*2!}{(1)! 2!} = \frac{3}{1}\) = 3 ways.

So, by the FUNDAMENTAL principle of counting we can SELECT a GROUP of 2 males and 2 females from the family in 10*3 = 30 ways.

Correct ANSWER is (c) 10

Explanation: Out of a group of 5 persons, we have to select 3 persons. This can be done in ^5C3 ways.

^nCr = \(\frac{n!}{(n-R)! r!}\)

^5C3 = \(\frac{5!}{(5-3)! 3!} = \frac{5*4*3!}{(2)! 3!} = \frac{20}{2}\)

i.e. 10 ways.

The CORRECT choice is (a) r!

The best I can explain: We know, ^NPR = \(\FRAC{n!}{(n-r)!}\) and ^nCr = \(\frac{n!}{(n-r)! r!}\).

=> ^nPr / ^nCr = \(\frac{\frac{n!}{(n-r)!}}{\frac{n!}{(n-r)! r!}}\) = r!

=> ^nPr = ^nCr * r!

Correct ANSWER is (b) False

To elaborate: Combination MEANS SELECTION and selection does not require ordering. So, ORDER does not MATTER in combination.

Correct option is (a) 4

To explain: If there are 4 paths to TRAVEL from Delhi to Kanpur and person take one path then in COMING BACK journey only one option is there. So, by the fundamental principle of counting, total NUMBER of ways a person can travel from Delhi to Kanpur and came back to Delhi via same path are 4*1 = 4.

The CORRECT ANSWER is (a) n!

The EXPLANATION: We know, ^NPR = \(\frac{n!}{(n-r)!}\).

^nPn = \(\frac{n!}{(n-n)!} = \frac{n!}{(0)!}\) = n!

Correct answer is (b) 1

The best I can EXPLAIN: We know, ^NCR = \(\frac{n!}{(n-R)! r!}\).

^nCn = \(\frac{n!}{(n-n)! n!} = \frac{n!}{(0)! n!}\) = 1/1 = 1.

The correct choice is (a) 27216

For EXPLANATION I would say: TOTAL DIGITS are 0-9 i.e.10. In a 5-digit number 0 cannot be at first place.

So, by the fundamental principle of counting, total numbers possible are 9*9*8*7*6 = 27216.

Correct answer is (c) 1344

Explanation: Total digits are 1-9 i.e.9. Even NUMBER means 2,4,6,8 at unit’s place.

So, 4 possible digits for unit place, after a digit is placed at unit place 8 digits are remaining.

8 ways to fill 1st POSITION, 7 ways to fill 2nd position, 6 ways to fill th3 3rd position.

Hence, by the fundamental principle of COUNTING, total possible numbers are 8*7*6*4 = 1344.

Correct choice is (b) 1

The explanation: We KNOW, ^nCr = \(\FRAC{n!}{(n-R)! r!}\).

^nC0 = \(\frac{n!}{(n-0)! 0!} = \frac{n!}{n!(1)}\) = 1.

The correct answer is (c) 625

The BEST EXPLANATION: We have to ARRANGE 4 letters of the 5 letters word IMAGE without repetition. So, TOTAL permutations are n^r = 5^4 = 625.

The correct CHOICE is (d) 2916

The best explanation: Total digits are 1-9 i.e.9. EVEN number means 2,4,6,8 at unit’s PLACE.

So, 4 possible digits for unit place. Rest all positions are FILLED in 9 ways.

Hence, by the fundamental principle of COUNTING, total possible numbers are 9*9*9*4 = 2916.

The correct option is (b) False

The BEST I can explain: PERMUTATION is known as arrangement. SELECTION is another name for combinations.

It INVOLVES arrangement of letters, numbers, persons ETC.

Right CHOICE is (a) True

The explanation: We know, ^NCR = \(\frac{N!}{(n-r)! r!}\).

REPLACING r by n-r, we get ^nCn-r = \(\frac{n!}{(n-(n-r))!(n-r)!} = \frac{n!}{r! (n-r)!}\) = ^nCr

=> ^nCr = ^nCn-r

Right option is (c) 6

Explanation: ^nP3 = 4*^nP2

\(\FRAC{n!}{(n-3)!} = 4 * \frac{n!}{(n-2)!}\)

=> n-2 = 4

=> n=6.

The CORRECT answer is (B) 1

The explanation: We know, ^nPr = \(\FRAC{N!}{(n-r)!}\).

^nP0 = \(\frac{n!}{(n-0)!} = \frac{n!}{(n)!}\) = 1.

The CORRECT answer is (b) 42

To explain I would say: We know, N! = n.(n-1).(n-2).(n-3)…… = n(n-1)!

\(\FRAC{7!}{5!} = \frac{7.6!}{5!} = \frac{7.6.5!}{5!}\) = 7.6 = 42.

The correct answer is (d) 2880

Explanation: There are 5 vowels in WORD EDUCATION. 5 vowels can be arranged in ^5P5 i.e. 5! ways. When all vowels are together, 5 vowels together FORM one LETTER and remaining 3 letters i.e. together 4 letters can be arranged in ^4P4 i.e. 4! ways. Total possible arrangements are 5! * 4! = 120*24 = 2880.

The correct answer is (a) 234

Easiest explanation: Since there are 26 LETTERS in English ALPHABET and 9 possible DIGITS from 1 to 9.

So, by the fundamental principle of counting, total possible codes are 26*9 = 234.

Right OPTION is (d) 3125

The BEST explanation: PULSE is a 5 LETTERS word. By the fundamental principle of counting, total NUMBER of possible words with repetition allowed are 5*5*5*5*5 = 3125.

The correct ANSWER is (a) 1

For explanation I WOULD SAY: We know, ^nCr + ^nCr-1 = ^n+1Cr

Substituting n=14 and r=4, we GET ^14Cr + ^14Cr-1 = ^15Cr

=>^14Cr-1 = ^15Cr – ^14Cr = 15-14 = 1.

Correct CHOICE is (c) 5

To elaborate: We know, ^NCP = ^nCq

=>either p = q or p + q = n.

Here p=2 and q=3 so, p ≠ q.

Hence p + q = n => n = p + q = 2 + 3 = 5.

Right choice is (a) 504

Easiest explanation: 1-9 digits which MEANS 9 digits are possible. We have to ARRANGE 3 digits at a time out of 9 digits WITHOUT repetition. So, TOTAL permutations are ^nPr = ^9P3 = \(\frac{9!}{(9-3)!} = \frac{9!}{6!}\) = 9.8.7 = 504.

Correct answer is (C) 99999

For EXPLANATION I would say: TOTAL digits are 0-9 i.e.10. In passcode, 0 may also OCCUR at first place.

All places are filled in 10 ways.

So, by the FUNDAMENTAL principle of counting, total numbers possible are 10*10*10*10*10=100000.

Maximum number of incorrect pass code entered = 100000-1 = 99999.

Correct answer is (b) 14

The BEST I can explain: We know, ^nCr = \(\frac{N!}{(n-r)! r!}\).

^2nC3 / ^nC3 = \(\frac{\frac{n!}{3! (2N-3)!}}{\frac{n!}{3! (n-3)!}}\)

= \(\frac{2n! (n-3)!}{n! (2n-3)!}\)

9/1 = \(\frac{2n(2n-1)(2n-2)}{n(n-1)(n-2)}\)

9 = \(\frac{2(2n-1)2}{(n-2)}\)

9n-18 = 8n-4

=>n=14.

Correct answer is (d) \(\FRAC{n!}{(n-R)! r!}\)

The best I can explain: Permutation is known as SELECTION. ^NCR means arranging r OBJECTS out of n.

^nCr = \(\frac{n!}{(n-r)! r!}\).

Right option is (d) 3125

The best I can EXPLAIN: IMAGE is a 5 letters WORD. We have to arrange all 5 letters of the word IMAGE with REPETITION allowed. So, total permutations are n^r = 5^5 = 3125.

Correct CHOICE is (c) 1663200

To explain: There are total 9 letters out of which 1T, 2N, 2D, 3E, 1P.

Total number of permutations are \(\FRAC{9!}{3!2!2!} = \frac{9.8.7.6.5.4.3.2.1}{6*2*2} = \frac{362880}{24}\) = 15120.

Correct option is (C) 625

To elaborate: PULSE is a 5 letters word. By the fundamental PRINCIPLE of counting, total number of possible WORDS with repetition ALLOWED are 5*5*5*5 = 625.

The correct choice is (c) 12

The best explanation: If there are 4 paths to TRAVEL from DELHI to Kanpur and person take one path then in coming back journey only 3 paths are remaining. So, by the fundamental principle of counting, total number of ways a person can travel from Delhi to Kanpur and came back to Delhi via different path are 4 * 3 = 12.

The correct CHOICE is (b) 729

To explain I WOULD SAY: 1-9 digits which means 9 digits are possible. We have to arrange 3 digits at a time out of 9 digits with repetition allowed. So, total PERMUTATIONS are n^r = 9^3 = 729.

Correct option is (a) 1

Easiest explanation: \(\frac{100}{10!} = \frac{1}{8!} + \frac{x}{9!}\).

We know, n! = n.(n-1). (n-2). (n-3) …………… = n(n-1)!

\(\frac{100}{10.9.8!} = \frac{1}{8!} + \frac{x}{9.8!}\)

=> \(\frac{100}{10.9} = \frac{1}{1} + \frac{x}{9}\)

=> \(\frac{10}{9} = 1 + \frac{x}{9}\)

=> \(\frac{x}{9} = \frac{10}{9} – 1 = \frac{1}{9}\)

=> x=1.

Correct ANSWER is (c) 120

Explanation: PULSE is a 5 letters word. By the FUNDAMENTAL principle of counting, TOTAL number of possible words without repetition are 5*4*3*2*1 = 120.

The correct choice is (d) 90000

For explanation: Total DIGITS are 0-9 i.e.10. In a 5-digit number 0 cannot be at FIRST place.

So, by the fundamental PRINCIPLE of COUNTING, total numbers POSSIBLE are 9*10*10*10*10=90000.

The correct option is (d) 16

The explanation is: If there are 4 paths to TRAVEL from DELHI to Kanpur then NUMBER of paths to travel from Kanpur to Delhi are 4. So, by the fundamental PRINCIPLE of counting, TOTAL number of ways to go and come back are 4*4 = 16.

Right answer is (c) 99

Easiest explanation: Words STARTING with letter A comes first in dictionary. Starting with A, number of words = 4! = 24. Starting with E, number of words = 4! = 24. Starting with I, number of words = 4! = 24. Starting with G, number of words = 4! = 24. SINCE our word also start with M so, we have to CONSIDER one more letter i.e. MA. Since our word also start with MA so, we have to consider one more letter i.e. MAE. Starting with MAE, number of words = 2! = 2. Since our word also start with MAG so, we have to consider one more letter i.e. MAGE. Starting with MAGE, only one letter i.e. MAGEI. After this, MAGIE comes. Total number of words before MAGIE = 24 + 24 + 24 + 24 + 2 = 98. So, rank of MAGIE is 99.

Correct answer is (B) 1260

To explain: Total NUMBER of pens are 2+3+4 = 9 out of which 2 are of 1 type, 3 are of 2^nd type and 4 are of 3^rd type so, total number of arrangements = \(\frac{9!}{2!3!4!} = \frac{9*8*7*6*5}{2*6}\) = 1260.

The correct answer is (c) 120

Easy explanation: PULSE is a 5 LETTERS word. By the fundamental principle of counting, total NUMBER of possible words without REPETITION are 5*4*3*2 = 120.

Right choice is (c) 120

The BEST I can EXPLAIN: We have to arrange 4 letters of the 5 letters word IMAGE without repetition. So, total PERMUTATIONS are ^NPR = ^5P4 = 5! / (5-4)! = 5! = 5.4.3.2.1 = 120.

Correct CHOICE is (b) 37440

To explain: There are 5 VOWELS in word EDUCATION. 5 vowels can be arranged in ^5P5 i.e. 5! WAYS. When all vowels are TOGETHER, 5 vowels together form one letter and remaining 3 letters i.e. together 4 letters can be arranged in ^4P4 i.e. 4! ways. Total possible arrangements are 5! * 4! = 120*24 = 2880.

So, when all vowels do not occur together, total possible arrangements = 8! – 2880 = 40320 – 2880 = 37440.