There are k-2 members in a certain band, including Jim and Allan. Two

Question

There are k-2 members in a certain band, including Jim and Allan. Two members are to be selected to attend the Grammy awards ceremony. If there are 10 possible combinations in which Jim and Allan are not selected, what is the value of k?

(A) 5
(B) 9
(C) 15
(D) 18
(E) 25

(A) 5
(B) 9
(C) 15
(D) 18
(E) 25

Bunuel · Accepted Answer

C^2_{(k-2-2)}=10 ;

C^2_{(k-4)}=10 ;

\frac{(k-4)!}{2!(k-6)!}=10 ;

\frac{(k-5)(k-4)}{2}=10  (reduced by (k-6)!);

(k-5)(k-4)=20 ;

k=9 .

Answer: B.

priyabaheti · Answer

any other way to solve this question?

ThatDudeKnows · Answer

priyabaheti

Let's just Plug In The Answers (PITA). I like trying B and D.

B: k=9, so there are 7 members in the band.
How many ways are there to select two people who aren't Jim or Allan? Let's call the other members P, Q, R, S, and T. There are five of them and we need two.
5C2 = (5*4)/(2*1) = 10 Yay, that's what we wanted!
Answer choice B.

PQ
PR
PS
PT
QR
QS
QT
RS
RT
ST

ThatDudeKnowsPITA

ScottTargetTestPrep · Answer

Since Jim and Allan are not selected, the two members to attend the awards ceremony were selected from among (k - 2) - 2 = k - 4 members. This choice can be made in  _{k - 4}C_2 = \frac{(k - 4)!}{2!	imes (k - 6)!}  ways. Recall that (k - 4)! = (k - 4)(k - 5)(k - 6)!. Let's substitute this expression for (k - 4)! in the numerator and simplify:

\Rightarrow\frac{(k - 4)!}{2!	imes (k - 6)!}

\Rightarrow\frac{(k - 4)(k - 5)(k - 6)!}{2!	imes (k - 6)!}

\Rightarrow\frac{(k - 4)(k - 5)}{2}

We are told that two members from among the members other than Jim and Allan can be chosen in 10 ways, so let's set the above expression equal to 10 and solve for k:

\Rightarrow\frac{(k - 4)(k - 5)}{2} = 10

\Rightarrow (k - 4)(k - 5) = 20

\Rightarrow k^2 - 5k - 4k + 20 = 20

\Rightarrow k^2 - 9k = 0

\Rightarrow k(k - 9) = 0

\Rightarrow k = 0\quad	ext{or}\quad k - 9 = 0

\Rightarrow k = 0\quad	ext{or}\quad k = 9

Notice that k cannot equal 0 because in that case, the band would contain k - 2 = -2 members, which is not possible. Thus, k must be equal to 9.

Answer: B

luisdicampo · Answer

Deconstructing the Question

Total members in the band:   k - 2  
  Members excluded from selection:  2 (Jim and Allan) 
  Available members to choose from:   (k - 2) - 2 = k - 4  
  Selection criteria:  Choosing 2 members out of  k - 4  results in  10 combinations .

Step 1: Set up the Combination Equation  
 (k - 4)C2 = 10 
 \frac{(k - 4)(k - 5)}{2} = 10 
 (k - 4)(k - 5) = 20

Step 2: Solve for k  
We are looking for two consecutive integers whose product is 20. Those numbers are  5 and 4 .
 k - 4 = 5 
 k = 9

Verification  
If  k = 9 , total members =  9 - 2 = 7 .
Excluding Jim and Allan, we have  7 - 2 = 5  members.
Ways to pick 2 from 5:  5C2 = (5 * 4) / 2 = 10 . (Matches the prompt).

The correct answer is B.

There are k-2 members in a certain band, including Jim and Allan. Two

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There are k-2 members in a certain band, including Jim and Allan. Two

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