Top 160 Permutation and Combination questions [ 100% FREE ]

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Top 160 Permutation and Combination questions:

131. A box contains 20 black and 15 pink balls. Two balls are picked from the box at random. Find the probability that they are of same colour.
एक बॉक्स में 20 काली और 15 गुलाबी गेंदें होती हैं। दो गेंदों को बॉक्स से यादृच्छिक रूप से उठाया जाता है। क्या प्रायिकता है कि वे एक ही रंग के हैं?A. 59/119
B. 54/119
C. 45/112
D. 89/112
E. None of these
Option “A” is correct.Required probability = Probability that both balls are black + Probability that both balls are pink

= [20C2+15C2]/35C2

= [190 + 105]/595

= 59/119

132. A box contains 4 blue balls, 5 pink balls and y green balls and the probability of getting a green ball is 1/4. If two balls are drawn at random from the box, then find the probability that all the balls are in same colour?
एक बॉक्स में 4 नीली गेंदें, 5 गुलाबी गेंदें और y हरी गेंदें हैं और हरी गेंद मिलने की प्रायिकता 1/4 है। यदि बॉक्स से यादृच्छिक रूप से दो गेंदें निकाली जाती हैं, तो सभी गेंदों के एक ही रंग के होने की प्रायिकता ज्ञात कीजिए?A. 19/66
B. 5/72
C. 21/92
D. 7/51
E. None of these
Option “A” is correct.The total number of balls in the box = 4 + 5 + y = 9 + y

yC1/(9 + y)C1 = 1/4

4y = 9 + y

3y = 9

y = 3

The number of green balls in the box = 3

The total number of balls in the box = 4 + 5 + 3 = 12

Required probability = (4C2 + 5C2 + 3C2)/12C2

= (6 + 10 + 3)/66

= 19/66

133. A box contains 4 pink balls, 6 grey balls, 4 brown balls. If three balls are drawn randomly from the bag, then what is the probability of all the balls being same colour?
एक बॉक्स में 4 गुलाबी गेंदें, 6 स्लेटी गेंदें, 4 भूरी गेंदें निहित हैं। यदि तीन गेंदों को थैले से यादृच्छिक ढंग से निकाला गया हैं, तो सभी गेंदों के समान रंग के होने की संभावना क्या है?A. 7/26
B. 4/13
C. 5/26
D. 2/13
E. 1/13
Option “E” is correct.Probability of choosing the balls of same colour,

Selected balls         (or)    selected balls          (or)    selected balls

is pink                    is grey                             is brown

Probability = 

134. A box contains 4 red balls, 6 green balls and 2 blue balls. If two balls are drawn at random from the box, then find the probability of that balls are same colour?
एक बॉक्स में 4 लाल गेंदें, 6 हरी गेंदें और 2 नीली गेंदें हैं। यदि बॉक्स से यादृच्छिक रूप से दो गेंदें निकाली जाती हैं, तो उन गेंदों के समान रंग के होने की प्रायिकता ज्ञात कीजिए?A. 3/10
B. 7/12
C. 2/5
D. 1/3
E. None of these
Option “D” is correct.The total number of balls in the box = 4 + 6 + 2 = 12

Required probability = (4C2 + 6C2 + 2C2)/12C2

= 1/3

135. A box contains 48 balls which is in three different colours(orange, green and gray) and the ratio of the number of orange and gray balls in the box is 5:7 respectively. If one ball is drawn randomly and the probability of getting a green ball is 1/4, then find the probability of selecting two same colour balls from the box?
एक बॉक्स में 48 गेंदें हैं जो तीन अलग-अलग रंगों (नारंगी, हरा और ग्रे) में हैं और बॉक्स में नारंगी और ग्रे गेंदों की संख्या का अनुपात क्रमशः 5:7 है। यदि एक गेंद यादृच्छया निकाली जाती है और हरी गेंद मिलने की प्रायिकता 1/4 है, तो बॉक्स में से दो समान रंग की गेंदों के चयन की प्रायिकता ज्ञात कीजिए?A. 127/376
B. 186/235
C. 115/336
D. 155/432
E. None of these
Option “A” is correct.Total number of balls=48

The number of green balls=x

xC1/48C1=1/4

x/48=1/4

x=12

The number of orange balls=(48-12)*5/12=15

The number of gray balls=36-15=21

Required probability=(15C2+12C2+21C2)/48C2

=127/376

136. A box contains 7 black, 6 white and 5 brown balls. 3 balls are drawn at random from the box. What is the probability that the balls drawn are not the same colour?
एक बॉक्स में 7 काली, 6 सफेद और 5 भूरी गेंदें हैं। बॉक्स से यादृच्छिक रूप से 3 गेंदें निकाली जाती हैं। इसकी क्या प्रायिकता है कि निकाली गई गेंदें एक ही रंग की न हों?A. 65/716
B. 21/127
C. 751/816
D. 545/816
E. None of these
Option “C” is correct.Total number of balls = 7 + 6 + 5 = 18

So, the 3 balls out of 18 balls can be drawn = 18c3

= (18 × 17 × 16)/(3 × 2)

= 816 ways

Now, if three balls are of the same colour,

= 7c3 + 6c3 + 5c3

= (7 × 6 × 5)/(3 × 2) + (6 × 5 × 4)/(3 × 2) + (5 × 4 × 3)/(3 × 2)

= 35 + 20 + 10 => 65 ways

So, the probability of that balls are not the same colour is,

= 1 –  (65/816)

= 751/816

137. A bag contains 6 green, some yellow and some blue balls. Probability of drawing one yellow ball from the bag is 1/3 and probability of drawing one green ball from the bag is 1/4. Find the probability of drawing three balls from the bag such that all the balls are of different colour.
एक बैग में 6 हरे, कुछ पीले और कुछ नीली गेंदें होती हैं। बैग से एक पीले रंग की गेंद को खींचने की संभावना 1/3 है और बैग से एक हरे रंग की गेंद को खींचने की संभावना 1/4 है। बैग से तीन गेंदों को खींचने की संभावना का पता लगाएं जैसे कि सभी गेंदें अलग-अलग रंग की हैं?A. 60/253
B. 57/259
C. 41/231
D. 47/247
E. None of these
Option “A” is correct.Number of green balls = 6

Let, number of yellow balls = x

And number of blue balls = y

Total number of balls = (6 + x + y)

x/(6 + x + y) = 1/3

=> 3x = 6 + x + y

=> 3x – x – y = 6

=> 2x – y = 6 ————— (i)

6/(6 + x + y) = 1/4

=> 24 = 6 + x + y

=> x + y = 24 – 6

=> x + y = 18 —————– (ii)

Adding (i) and (ii)

2x – y + x + y = 6 + 18

=> 3x = 24

=> x = 24/3

=> x = 8

From (ii)

8 + y = 18

=> y = 18 – 8

=> y = 10

Total number of balls = 6 + 8 + 10 = 24

Required probability = (6c1 x 8c1 x 10c1)/24c3

= (6 x 8 x 10)/2024

= 60/253

138. A bag contains four different coloured balls viz. red, green, yellow and blue. A child picked a ball from the bag. The child wants a ball of one among the three colour viz. green, red and yellow. Probability of getting a green ball is 0.15, probability of getting a red ball is 0.25 and probability of getting blue ball is 0.3. Find the probability that the child will get a ball of a colour of his choice.
एक बैग में चार अलग-अलग रंगीन गेंदें होती हैं। लाल, हरा, पीला और नीला। एक बच्चे ने बैग से एक गेंद निकाली। बच्चा तीन रंगों में से एक की एक गेंद हरा, लाल और पीला चाहता है। हरे रंग की गेंद मिलने की संभावना 0.15 है, लाल गेंद मिलने की संभावना 0.25 है और नीली गेंद मिलने की संभावना 0.3 है। इस संभावना को ढूंढें कि बच्चे को उसकी पसंद के रंग की एक गेंद मिलेगी।A. 2/5
B. 3/10
C. 7/10
D. 3/5
E. None of these
Option “C” is correct.Probability of getting a yellow ball = 1 – 0.15 – 0.25 – 0.3 = 0.3

Required probability = 0.15 + 0.25 + 0.3 = 0.7 = 7/10

139. A box contains 5 pink, 6 blue and 7 brown marbles. 3 marbles are picked up randomly from the box. Then find the probability that the marbles drawn contains marbles of different colours?
एक बॉक्स में 5 गुलाबी, 6 नीले और 7 भूरे रंग के मार्बल्स होते हैं। 3 मार्बल्स को बॉक्स से यादृक्षिक रूप से उठाया जाता है। फिर निकाले गए मार्बल्स में अलग-अलग रंगों के मार्बल्स होने के क्या प्रायिकता हैं?A. 10/101
B. 11/121
C. 24/125
D. 35/136
E. None of these
Option “D” is correct.Total number of marbles is = 5 + 6 + 7 = 18

So, the total number of ways = 18c3

= (18 × 17 × 16)/(3 × 2)

= 816

Now, to have 3 different colored marbles, the selection of one marble of each colour is to be made,

= 5c1 × 6c1 × 7c1

= 5 × 6 × 7

= 210

Thus, the required probability is = 210/816 => 35/136

140. A box contains 5 yellow balls, 3 red balls and 6 white balls. If three balls are drawn out randomly, find the probability that the balls are of different colours?
एक बॉक्स में 5 पीले रंग की गेंदें, 3 लाल रंग की गेंदें और 6 सफेद रंग की गेंदें हैं। यदि तीन गेंदों को यादृच्छिक ढंग से निकाला गया हैं, तो संभावना ज्ञात कीजिये कि गेंदें अलग-अलग रंगों की हैं?A. 56 / 177
B. 45 / 182
C. 38 / 195
D. 62 / 213
E. None of these
Option “B” is correct.Total balls = 5 + 3 + 6 = 14

n (E) = 5C1 and 3C1 and 6C1

n (S) = 14C3

P (E) = n (E) / n (S)

P (E) = (5C1 and 3C1 and 6C1) / (14C3)

P (E) = 45 / 182