**Find the sum of all the 4 digit numbers that can be formed with the digits 3, 4, 5 and 6**1. 1199882. 119883. 1919884. None of theseAnswer : Option 1

No. of Digits = 4 All are distinct; They can be arranged in 4! = 24 ways

Each of the digits 3, 4, 5 and 6 occur at unit place = 3! Ways = 6 ways.

Thus there will be 6 numbers ending with 3, 4, 5 and 6 each. So the sum of the digits at unit's place = 6(3 + 4 + 5 + 6) =108

The sum of numbers = 108 × 10^{3}+ 108 × 10^{2}+ 108 × 10^{1}+ 108 × 10^{0}= 119988**Find the sum of all the 4 digit numbers that can be formed with the digits 3, 4, 4 and 2.**1. 433392. 439993. 433294. None of theseAnswer : Option 3

Here each of the digits 2 and 3 will occur at unit, tens, hundred and thousand place (^{3}P_{3}/2!) = 3 times. Digit 4 will occur at each place = 6 times;

∴ Sum of digits at unit, tens, hundred and thousand place = 3 × 3 + 6 × 4 + 3 × 2 =39. Sum of numbers formed =

= 39 × 10^{3}+ 39× 10^{2}+ 39 × 10^{1}+ 39 × 10^{0}= 43329**The number of straight lines that can be drawn out of 12 points of which 8 are collinear is**1. 392. 293. 494. 59Answer : Option 1

The required number of lines=^{12}C_{2}–^{8}C_{2}+ 1 = 1 + 66 – 28 = 39**A box contains three white balls, four black balls and three red balls. The number of ways in which three balls can be drawn from the box so that at least one of the balls is black is**1. 502. 1003. 1504. 200Answer : Option 2

The required number of ways

(a) 1 black and 2 others =^{4}C_{1}.^{6}C_{2}= 4 × 15 = 60

(b) 2 black and 1 other =^{4}C_{2}.^{6}C_{1}= 6 × 6 = 36

(c) All the three black =^{4}C_{3}= 4

Total =60 + 36 + 4 = 100**In a cricket championship, there are 21 matches. If each team plays one match with every other team, the number of teams is**1. 72. 93. 104. None of theseAnswer : Option 1

Let n be the number of teams.

^{n}C_{2}= 21

(n(n-1)/2) = 21

⇒ n(n-1) = 42 ∴

⇒ n = 7

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**The number of four–digit telephone numbers having at least one of their digits repeated is**1. 9,0002. 1,00,003. 3,2404. 4960Answer : Option 4

The number of four–digit telephone numbers which can be formed using the digits of 0, 1, 2,...., 9 is 10^{4}.

The number of four digit telephone numbers which have none of their digits repeated is^{10}P_{4}= 10 ×9 ×8 ×7 = 5040

Hence the required number =10^{4}- 5040 = 4960**In how many ways can you rearrange the word JUMBLE such that the rearranged word starts with a vowel?**1. 1202. 2403. 3604. 60Answer : Option 2

JUMBLE is a six-lettered word. Since the rearranged word has to start with a vowel, the first letter can be either U or E. The balance 5 letters can be arranged in^{5}P_{5}or 5! ways. Total number of words = 2 × 5! = 240.**In an examination, a candidate is required to pass all five different subjects. The number of ways he can fail is:**1. 322. 313. 304. 29Suggested Action:Answer : Option 2

The candidate will fail if he fails either in 1 or 2 or 3 or 4 or 5 subjects, ∴ Required number of ways^{5}C_{1}+^{5}C_{2}+^{5}C_{3}+^{5}C_{4}+^{5}C_{5}= 31**The sum of the divisors of 2**^{6}.3^{4}.5^{7}is1. 2^{6}.3^{8}.5^{4}.7^{3}2. 2^{7}.3^{5}.5^{8}-2.3.53. 2^{6}.3^{8}.5^{4}.7^{3}-14. None of theseAnswer : Option 4

Any divisor of 2^{6}.3^{4}.5^{7}. is of the form 2^{a}.3^{b}.5^{c}. where 0≤ a ≤ 6, 0 ≤ b ≤ 4 & 0 ≤ c ≤ 7.

Thus the sum of the divisors of 2^{6}.3^{4}.5^{7}.

is (1 + 2+…..+2^{5}+ 2^{6})(1 + 3 + …. + 3^{4})(1 + 5 +....+ 5^{7}) =

((2^{7}-1)/(2-1))((3^{5}-1)/(3-1))((5^{8}-1)/(5-1))

=((2^{7}-1)(3^{5}-1)(5^{8}-1)/2.4)**Nine chairs are numbered 1 to 9. Three women and four men wish to occupy one chair each. First the women chose the chairs from amongst the chair marked 1 to 5; and then the men select the chairs from amongst the remaining. The number of possible arrangements is**1.^{5}C_{3}×^{4}C_{2}2.^{5}C_{2}×^{4}P_{3}3.^{5}C_{3}×^{6}C_{4}4. None of theseAnswer : Option 3

Women can select 3 chairs from chairs numbered 1 to 5 in^{5}C_{3}ways and remaining 6 chairs can be selected by 4 men in^{6}C_{4}ways. Hence the required number of ways =^{5}C_{3}×^{6}C_{4}