Draupadi has 4 friends in how many ways can she invite one or more of them to a dinner

Explanation Option [a] is correct. She may invite one or more friends by selecting either 1 or 2 or 3 or 4 or 5 friends out of 5 friends. 1 friend can be selected out of 5 in ways 2 friends can be selected out of 5 in ways 3 friends can be selected out of 5 in ways 4 friends can be selected out of 5 in ways 5 friends can be selected out of 5 in ways Hence the required number of ways Alternatively: since,

• Correct Answer: A

  Solution :

  Explanation Option [a] is correct. She may invite one or more friends by selecting either 1 or 2 or 3 or 4 or 5 friends out of 5 friends. \[\therefore \] 1 friend can be selected out of 5 in \[^{5}{{C}_{1}}\] ways 2 friends can be selected out of 5 in \[^{5}{{C}_{2}}\] ways 3 friends can be selected out of 5 in \[^{5}{{C}_{3}}\] ways 4 friends can be selected out of 5 in \[^{5}{{C}_{4}}\] ways 5 friends can be selected out of 5 in \[^{5}{{C}_{5}}\] ways Hence the required number of ways \[{{=}^{5}}{{C}_{1}}{{+}^{5}}{{C}_{2}}{{+}^{5}}{{C}_{3}}{{+}^{5}}{{C}_{4}}{{+}^{5}}{{C}_{5}}\] \[=5+10+10+5+1=31\] Alternatively: \[^{5}{{C}_{1}}+{{\,}^{5}}{{C}_{2}}+{{\,}^{5}}{{C}_{3}}+{{\,}^{5}}{{C}_{4}}+{{\,}^{5}}{{C}_{5}}={{2}^{5}}-1=31\] since,\[^{n}{{C}_{1}}+{{\,}^{n}}{{C}_{2}}+{{\,}^{n}}{{C}_{3}}+\,.....+{{\,}^{n}}{{C}_{n}}={{2}^{n}}-1\]                  

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Page 2

• Correct Answer: A

  Solution :

  Explanation Option [a] is correct. A committee of 5 persons, consisting of at most two ladies, can be formed can the following ways.

  (i) selecting 5 gents only out of 6.

  (ii) selecting 4 gents only out of 6 and one lady out of 4.

  (iii) selecting 3 gents only out of 6 and two ladies out of 4.

  In case I, the number of ways \[={{\,}^{6}}{{C}_{5}}\] In case II, the number of ways \[={{\,}^{6}}{{C}_{4}}\times {{\,}^{4}}{{C}_{2}}\] In case III, the number of ways \[={{\,}^{6}}{{C}_{5}}+{{\,}^{6}}{{C}_{4}}\times {{\,}^{4}}{{C}_{1}}+{{\,}^{6}}{{C}_{3}}\times {{\,}^{4}}{{C}_{2}}\] \[\therefore \] Required number of ways \[=6+60+120=186\]              

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Page 3

• Correct Answer: A

  Solution :

  Explanation Option [a] is correct. There are 11 letters in the given word of which 2 are A,s, 2 are I's 2 are N?s and the remaining 5 letters are different. Thus we have 11 letters of 8 different kinds viz., \[\left( A, A \right), \left( I, I \right), \left( N, N \right),\] E, X, M, T, 0. A group of 4 letters can be classified as follows:

  (i) Two alike of one kind and two alike of another kind.

  (ii) Two alike and the other two different.

  (iii) All four different.

  In case I, the number of ways \[={{\,}^{3}}{{C}_{2}}=3\] Incase II, the number of ways \[={{\,}^{3}}{{C}_{1}}\times {{\,}^{7}}{{C}_{2}}=63\] In case III, the number of ways \[={{\,}^{8}}{{C}_{4}}=63\] Hence, the required number of ways \[=3+63+70=136\]               

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Lisa has 5 friends in how many ways can she invite one or more of them [#permalink]

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Lisa has 5 friends in how many ways can she invite one or more of them at a dinner party?(A) 63(B) 32(C) 31(D) 25

(E) 24

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Re: Lisa has 5 friends in how many ways can she invite one or more of them [#permalink]

Bunuel wrote:

Lisa has 5 friends in how many ways can she invite one or more of them at a dinner party?(A) 63(B) 32(C) 31(D) 25

(E) 24

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Re: Lisa has 5 friends in how many ways can she invite one or more of them [#permalink]

Bunuel wrote:

Lisa has 5 friends in how many ways can she invite one or more of them at a dinner party?(A) 63(B) 32(C) 31(D) 25

(E) 24

Alternate Solution:

Answer: C

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Re: Lisa has 5 friends in how many ways can she invite one or more of them [#permalink]

Given• Lisa has 5 friends.

To find

Approach and Working out:

Method 1

• Total ways = 5C1 +5c2 +5c3+5c4 +5c5

Correct Answer: Option C

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Re: Lisa has 5 friends in how many ways can she invite one or more of them [#permalink]

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Re: Lisa has 5 friends in how many ways can she invite one or more of them [#permalink]