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10 business executives and 7 chairmen meet at a conference. If each business executive shakes the hand of every other business executive and every chairman once, and each chairman shakes the hand of each of the business executives but not the other chairmen, how many handshakes would take place? (A) 144 (B) 131 (C) 115 (D) 90 (E) 45 Source: GMAT Club Tests - hardest GMAT questions Chairmen shake hands 10*7=70 with business executives times. Business executives shake hands with each other 10 C 2 times or 45 times. The total is 115 . Can someone explain to me why its 10 C 2?
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bipolarbear wrote: 10 business executives and 7 chairmen meet at a conference. If each business executive shakes the hand of every other business executive and every chairman once, and each chairman shakes the hand of each of the business executives but not the other chairmen, how many handshakes would take place?
144
131
115
90
45
Can someone explain to me why its 10 C 2? 10 business executives shakes hands with other 9 business executives in 10c2 ways = 45 ways 10c2 = 9+8+7+6+5+4+3+2+1 = 45 First executive shakes hands with remaining 9 executives Second executive shakes hands with remaining 8 executives Third executive shakes hands with remaining 7 executives Fourth executive shakes hands with remaining 6 executives Fifth executive shakes hands with remaining 5 executives Sixth executive shakes hands with remaining 4 executives Seventh executive shakes hands with remaining 3 executives Eighth executive shakes hands with remaining 2 executives Nineth executive shakes hands with remaining 1 executives Tenth executive already shakes hands with all 9 executives. 7 chairmen each shake hands with all 10 executive in 10x7 - 70 ways Total handshakes = 45+70 = 115
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Another method 17 c 2 - 7 c 2 = 115 17 c 2 = total number of hand shakes between 17 people 7 c 2 = total number of hand shakes between chairmen
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Here's how I did it:
Scenario 1: business shakes hand only with another businessman only
#ways to select 1 businessman = 10
#ways to select another businessman = 9
#handshakes between businessmen only = 10x9/2 = 45 (We divide by 2 since order does not matter)
Scenario 2: businessman shakes hand with a chairman only
#was to select 1 businessman = 10
#ways to select 1 chairman = 7
#handshakes between businessmen and chairmen = 10x7/2 = 35
Scenario 3: chairman shakes hand with a businessman only.
#ways to select 1 chairman = 7
# was to select a businessman = 10
#handshakes between chairmen only = 7x10/2 = 35
Total handshakes = 45 + 35 + 35 = 115
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Lets think of it this way. It takes 2 to tango, or in this case shake hands The total number of shakehand greetings = The total Nr of 2 person groups that we can form. 10 Execs Nr of 2 person groups we can form = 10 C 2 = 45 7 Chairman and 10 Execs Nr of 2 person groups we can form, which INCLUDE 1 chairman and 1 Exec 7 C 1 x 10 C 1 = 70 .. Total 70 + 45 = 115 Here is an easy way to calculate permutations and combinations without using the formula
n P M = n x n-1 x n-2 ...m times
10 P 3 = 10 x 9 x 8 ( i.e 3 terms)
11 P 6 = 11 x 10 x 9 x 8 x 7 x 6 (i.e 6 terms)n C m = (n x n-1 x n-2 ...m times) / 1 x 2 .. m terms
10 C 2 = 10 x 9 / 1 x 2
11 C 8 = 11 C 3 = 11 x 10 x 9 / 1 x 2 x 3
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Oh... thank you. I read the question as "shakes hands with every other business executive" as shaking hands with alternating executives 2, 4, 6, 8, 10... which clearly did not make sense. +1 for you
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here it is :
It takes 2 persons for one hand shake.
there are total 10 E + 7 C = 17 persons.
so total number of handshakes possible is 17C2.
The question is trying to confuse you by putting two different conditions.
17 C2 is possible when all people shakes hand with each other. But from the second condition , no chairman shakes hand with other chairman but executive.. so 7 persons are not shaking hands among themselves. So 7C2 cases must be subtracted.
Final Answer 17C2 - 7C2 = 115
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Here is my version of explanation:
If we have 3 people (Ex: P1, P2, P3), then the possible different shakehands are:
(P1, P2)(P1,P3) (P2,P3)
If we have 4 people (Ex: P1, P2, P3, P4), then the possible different shakehands are:
(P1, P2)(P1,P3) (P1,P4) (P2, P3) (P2, P4) (P3, P4) = Choosing 2 from 4, i.e, 4C2
If each business executive shakes the hand of every other business executive ... possible ways for this scenario are: 10C2=45
and every chairman once ... possible ways for this scenario are: 10 * 7 = 70
and each chairman shakes the hand of each of the business executives but not the other chairmen ....This is already covered as part of 10C2
So, Total: 45 + 70 = 115
Cheers!
Ravi
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I thought of it a different way... First i broke up the people into 2 groups execs and chairmen first the chairmen they each shake the 10 execs hands only.....10x7=70 next the execs: they shake in sets of 2 10C2 (taking out permutations) = 45 45+70 = 115 Best
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Its C.
10 Executives 7 Chairman
10 executives among themselves shake hands 9+8+7+6+5+4+3+2+1 = 45 times
7 chairman shakes hands with 10 executives 7*10 = 70 times
So total handshakes = 45 + 70 = 115
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10C2 +10x7=45+70=115
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The answer is C. as per the above mentioned problem, here is the reasoning that will follow: Let's assume each business exec. is assigned an alphabet A thru J Let's assume each chairman is assigned a number 1 thru 7. Now, starting with A (counting all busi. execs. and chairmen), A would shake 16 unique hands in all. Next, proceeding to B (counting all busi. execs. and chairmen), B would shake 15 unique hands in all (A would be excluded). Similarly, calculate this for C, D, E, F, G, H, I and J. Since the chairmen do not exchange handshakes among themselves, this isn't required. Now add all the unique handshakes to find the total: 16+15+14+13+12+11+10+9+8+7 = 115
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70 handshakes from Directors to Executive
10C2 for handshakes with Exe to Exe = 45 Total Handshakes = 70+45 = 115
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bipolarbear wrote: Oh... thank you. I read the question as "shakes hands with every other business executive" as shaking hands with alternating executives 2, 4, 6, 8, 10... which clearly did not make sense. +1 for you That was exactly how I interpreted the question. Now I get it. 
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bipolarbear wrote: 10 business executives and 7 chairmen meet at a conference. If each business executive shakes the hand of every other business executive and every chairman once, and each chairman shakes the hand of each of the business executives but not the other chairmen, how many handshakes would take place? (A) 144 (B) 131 (C) 115 (D) 90 (E) 45 Source: GMAT Club Tests - hardest GMAT questions Chairmen shake hands 10*7=70 with business executives times. Business executives shake hands with each other 10 C 2 times or 45 times. The total is 115 . Can someone explain to me why its 10 C 2? Total # of handshakes possible between 10+7=17 people (with no restrictions) is # of different groups of two we can pick from these 10+7=17 people (one handshake per pair), so C^2_{17}. The same way: # of handshakes between chairmen C^2_{7} (restriction). Desired=Total-restriction=C^2_{17}-C^2_{7}=115. Approach #2:Direct way: # of handshakes between executives C^2_{10} plus 10*7 (as each executive shakes the hand of each 7 chairmen): C^2_{10}+10*7=115. Answer: C.
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My take: There are 10 executives and 7 chairman.. The given condition is that each executive can shake with another person only once. That stands to (10*9)/2 +(10*7)/2 which equals 80. I have divided the above by 2 to reduce redundancy, i.e., A shakes with B is the same thing as B shakes with A. Moreover, each chair doesn't shakes with other chairman but the executives. Hence (7*10)/2=35. Total=80+35=115. C
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10c2+10*7 very common kind of question
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It took me a significantly longer time than 2 mins and in the end I kind of guessed C. Honestly it was more of just a guess and not an educated guess. Going through the explanations, it has helped me realize where I was going wrong. I was considering the total handshakes by a business executive as (9+7) and was summing this for a total of 7 executives. The explanations above really helped me understand my mistake. (Total handshakes - between chairmen) was a real smart way to go about the solution.
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