Permutation And Combination - Study Mode

[#66] A committee of 5 members is to be formed out of 3 trainees, 4 professors and 6 research associate. In how many different ways can this be done if the committee should have 2 trainees and 3 research associates?
Correct Answer

(C) 60

Explanation

Solution: Required number of ways $$eqalign{
& = left( {{}^3{C_2} imes {}^6{C_3}}
ight) cr
& = left( {{}^3{C_1} imes {}^6{C_3}}
ight) cr
& = left( {3 imes frac{{6 imes 5 imes 4}}{{3 imes 2 imes 1}}}
ight) cr
& = 60 cr} $$

[#67] In how many different ways can the letters of the word CREAM be arranged?
Correct Answer

(B) 120

Explanation

Solution: The given word contains 5 letters, all different. ∴ Required number of ways $$eqalign{
& = 5! cr
& = left( {5 imes 4 imes 3 imes 2 imes 1}
ight) cr
& = 120 cr} $$

[#68] In how many different ways can the letters of the word ALLAHABAD be arranged?
Correct Answer

(C) 7560

Explanation

Solution: The given word contains 9 letters, namely 4A, 2L, 1H, 1B and 1D. ∴ Required number of ways $$eqalign{
& = frac{{9!}}{{4! ,2! ,1! ,1! ,1!}} cr
& = frac{{9 imes 8 imes 7 imes 6 imes 5 imes 4 imes 3 imes 2 imes 1}}{{4 imes 3 imes 2 imes 1 imes 2}} cr
& = 7560 cr} $$

[#69] From a group of 7 men 6 women, 5 persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done?
Correct Answer

(D) 756

Explanation

Solution: Required number of ways $$ = left( {{}^7{C_3} imes {}^6{C_2}}
ight) + $$ xa0 $$left( {{}^7{C_4} imes {}^6{C_1}}
ight) + $$ xa0 $$left( {{}^7{C_5} imes {}^6{C_0}}
ight)$$ $$ = left{ {frac{{7 imes 6 imes 5}}{{3!}} imes frac{{6 imes 5}}{{2!}}}
ight}$$ xa0 xa0 $$ + left( {{}^7{C_3} imes {}^6{C_1}}
ight)$$ xa0 $$ + left( {{}^7{C_2} imes 1}
ight)$$ $$ = left{ {frac{{7 imes 6 imes 5}}{6} imes frac{{6 imes 5}}{{2 imes 1}}}
ight}$$ xa0 xa0 $$ + left( {frac{{7 imes 6 imes 5}}{{3 imes 2 imes 1}} imes 6}
ight)$$ xa0xa0 $$ + left( {frac{{7 imes 6}}{{2 imes 1}} imes 1}
ight)$$ $$eqalign{
& = left( {525 + 210 + 21}
ight) cr
& = 756 cr} $$

[#70] In how many different ways can the letters of the word BANKING be arranged in such a way that the vowels always come together?
Correct Answer

120

Explanation

Solution: The given words contains 7 letters of which N is taken 2 times. We keep the vowels (AI) together and treat them as 1 letter. Thus, we have to arrange 6 letters BNKNG (AI) of which N occurs 2 times and rest are different. These can be arranged in $$frac{6!}{2!}$$ = (6 × 5 × 4 × 3 ) = 360 days Now 2 vowels (AI) can be arranged among themselves in 2 ways. ∴ Required number of ways = (360 × 2) = 720