Question

There are 3 indians and 3 chinese in a group of 6 people.How many subgroups of this we choose so that every subgroup has atleast one indian

Answer 1

Suppose in a group which consist of 6 people(3 Indian + 3 Chinese):

  is represented as $(I_{1},I_{2},I_{3}),(C_{1},C_{2},C_{3})$

Subgroup means subsets of that Group.

Prerequisite Condition: There should be atleast one indian in each Group.

Number of Subgroups taking one Indian at a time = 3→$(I_{1}),(I_{2}),(I_{3})$

Number of subgroups taking two Indian at a time = 3→→$(I_{1},I_{2}),(,(I_{2},I_{3}), (I_{1},I_{3})$

Number of subgroups taking three Indian at a time = 1→→$(I_{1},I_{2},I_{3})$

Number of subgroup taking one Chinese and one Indian at a time = 9

Number of subgroup taking two Chinese and one Indian at a time = 9

Number of subgroup taking three Chinese and one Indian at a time = 3

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Number of subgroup taking three Chinese and one Indian at a time = 3→→$(I_{1},C_{1},C_{2},C_{3})(I_{2},C_{1},C_{2},C_{3}),(I_{3},C_{1},C_{2},C_{3})$

Number of subgroup taking three Chinese and two Indian at a time = 3→→$[(I_{1},I_{2},C_{1},C_{2},C_{3}),(I_{2},I_{3},C_{1},C_{2},C_{3})(I_{1},I_{3},C_{1},C_{2},C_{3})]$

Number of subgroup taking three Chinese and three Indian at a time = 1→→ $(I_{1},I_{2},I_{3}),(C_{1},C_{2},C_{3})$

Number of subgroup taking three Indian and one Chinese at a time = 3→→ $(I_{1},I_{2},I_{3},C_{1}),(I_{1},I_{2},I_{3},C_{2}),(I_{1},I_{2},I_{3},C_{3})$

Number of subgroup taking two Chinese and three Indian at a time = 3→→ $(I_{1},I_{2},I_{3},C_{1},C_{2})(I_{1},I_{2},I_{3},C_{1},C_{3}),(I_{1},I_{2},I_{3},C_{3},C_{2})$

Total number of subgroups which has six people (3 Chinese + 3 Indians) consisting atleast one Indian in each group=

  $_{1}^{3}\textrm{C}+_{2}^{3}\textrm{C}+_{3}^{3}\textrm{C}+_{1}^{3}\textrm{C}_{1}^{3}\textrm{C}+_{1}^{3}\textrm{C}_{2}^{3}\textrm{C}+_{1}^{3}\textrm{C}_{3}^{3}\textrm{C}+_{2}^{3}\textrm{C}_{1}^{3}\textrm{C}+_{2}^{3}\textrm{C}_{2}^{3}\textrm{C}+_{2}^{3}\textrm{C}_{3}^{3}\textrm{C}+_{3}^{3}\textrm{C}_{1}^{3}\textrm{C}+_{3}^{3}\textrm{C}_{2}^{3}\textrm{C}+_{3}^{3}\textrm{C}_{3}^{3}\textrm{C}=3+3+1+9+9+3+9+9+3+3+3+1=56$

Where, n(C,r)= $\frac{n!}{(n-r)!r!}$

SO, total number of subgroups= 56

Answer 2

Answer:

Step-by-step explanation:

Number of subgroups possible..2^6-1

Number of subgroups with no Indian..2^3-1.

Subtracting those two we get number of subgroups with atleast one Indian..

63-7= 56