23.
A family consisting of one mother, one father, two daughters and a son is taking a road trip in a sedan. The sedan has two front seats and three back seats. If one of the parents must drive and the two daughters refuse to sit next to each other, how many possible seating arrangements are there?
(A)28 (B)32 (C)48 (D)60 (E)120
Solution:
Since either one of the parents must drive, a parent can be selected in 2C1 = 2ways Since the 2 daughters refuse to sit together, figure out the separate ways the daughters do NOT sit together. Start with the back-seat: Both daughters in the back seat (near each window) can be arranged in 2ways. The middle place can be filled either by the Son or the other parent. This can be done in 2C1 = 2ways.
So total no: of ways such that both daughters sit in the back = 2x2x2 = 8 Front Seat: Pick 1 daughter to sit in the front seat: 2C1 = 2 The parent, son and daughter can now be arranged in the back seat in 3! ways. Total no: of ways such that 1 daughter sits in the front and the other daughter sits in the back = 2x2x6 = 24 Therefore, total number of seating arrangements = 24+8 = 32
24.
If {n} denote the remainder when 3n is divided by 2 then which of the following is equal to 1 for all positive integers n? I. {2n+1} II. {2n}+1 III. 2{n+1} A. I only B. II only C. I and II D. III only E. II and III
Solution:
Here, we're given this new operation {}. We're told that {n} equals the remainder when 3n is divided by 2. It's often helpful to do a couple of quick concrete examples to help you understand the operation. So: {2} = rem of (3*2/2) = rem of 6/2 = 0 {3} = rem of (3*3/2) = rem of 9/2 = 1 We can quickly determine that if n is odd, the remainder will be 1; if n is even, the remainder will be 0. Accordingly: {odd} = 1 {even} = 0 Now to the exact question: which of the following is equal to 1 for all positive integers n? We can quickly eliminate (III), since 2*(integer) is never going to equal 1 (even if we didn't understand the operation, we should be able to cross out III). III isn't part of the solution: eliminate (D) and (E). The other two statements occur with equal frequency in A/B/C, so let's start with I, which looks a bit simpler: I {2n+1} Well, we know that 2n will be even, so 2n+1 will be odd. Based on our previous work, we know that {odd}=1, so (I) satisfies the question: eliminate B (only A and C left!).
II {2n} + 1 Again, we know that 2n will be even and that {even}=0. So, II is really: 0 + 1 = 1 and, consequently, II also satisfies the question. I and II are both always equal to 1: choose C.
A family consisting of one mother, one father, two daughters and a son is taking a road trip in a sedan. The sedan has two front seats and three back seats. If one of the parents must drive and the two daughters refuse to sit next to each other, how many possible seating arrangements are there?
(A)28 (B)32 (C)48 (D)60 (E)120
Solution:
Since either one of the parents must drive, a parent can be selected in 2C1 = 2ways Since the 2 daughters refuse to sit together, figure out the separate ways the daughters do NOT sit together. Start with the back-seat: Both daughters in the back seat (near each window) can be arranged in 2ways. The middle place can be filled either by the Son or the other parent. This can be done in 2C1 = 2ways.
So total no: of ways such that both daughters sit in the back = 2x2x2 = 8 Front Seat: Pick 1 daughter to sit in the front seat: 2C1 = 2 The parent, son and daughter can now be arranged in the back seat in 3! ways. Total no: of ways such that 1 daughter sits in the front and the other daughter sits in the back = 2x2x6 = 24 Therefore, total number of seating arrangements = 24+8 = 32
24.
If {n} denote the remainder when 3n is divided by 2 then which of the following is equal to 1 for all positive integers n? I. {2n+1} II. {2n}+1 III. 2{n+1} A. I only B. II only C. I and II D. III only E. II and III
Solution:
Here, we're given this new operation {}. We're told that {n} equals the remainder when 3n is divided by 2. It's often helpful to do a couple of quick concrete examples to help you understand the operation. So: {2} = rem of (3*2/2) = rem of 6/2 = 0 {3} = rem of (3*3/2) = rem of 9/2 = 1 We can quickly determine that if n is odd, the remainder will be 1; if n is even, the remainder will be 0. Accordingly: {odd} = 1 {even} = 0 Now to the exact question: which of the following is equal to 1 for all positive integers n? We can quickly eliminate (III), since 2*(integer) is never going to equal 1 (even if we didn't understand the operation, we should be able to cross out III). III isn't part of the solution: eliminate (D) and (E). The other two statements occur with equal frequency in A/B/C, so let's start with I, which looks a bit simpler: I {2n+1} Well, we know that 2n will be even, so 2n+1 will be odd. Based on our previous work, we know that {odd}=1, so (I) satisfies the question: eliminate B (only A and C left!).
II {2n} + 1 Again, we know that 2n will be even and that {even}=0. So, II is really: 0 + 1 = 1 and, consequently, II also satisfies the question. I and II are both always equal to 1: choose C.
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