MAH MBA CET 2026 April 7 Shift 2 Question Paper: Download Answer Key with Solutions

Step 1: Understanding the Concept:

When a specific item or person must always be included in a selection, that person's place is fixed. This effectively reduces both the total number of items to choose from and the number of spots remaining to be filled.


Step 2: Key Formula or Approach:

If 1 particular person is always included, the number of ways to choose \(r\) people from \(n\) is: \[ ^{n-1}C_{r-1} \]


Step 3: Detailed Explanation:

1. Total number of persons (\(n\)) = 10.
2. Selection size (\(r\)) = 4.
3. Since 1 particular person is always included, we have already picked 1 person.
4. Remaining persons to choose from = \(10 - 1 = 9\).
5. Remaining spots to fill = \(4 - 1 = 3\).
6. Number of ways = \(^9C_3\). \[ ^9C_3 = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} \] \[ = 3 \times 4 \times 7 = 84. \]


Step 4: Final Answer:

The number of ways to make the selection is 84. Quick Tip: "Always included" \(\implies\) Subtract from both \(n\) and \(r\).
"Always excluded" \(\implies\) Subtract only from \(n\).

Step 1: Understanding the Concept:

To form a rectangle, we need to select 2 horizontal lines (to form the top and bottom) and 2 vertical lines (to form the left and right sides).


Step 2: Key Formula or Approach:

If there are \(m\) horizontal lines and \(n\) vertical lines, the number of rectangles is: \[ ^mC_2 \times ^nC_2 \]


Step 3: Detailed Explanation:

1. Number of horizontal lines (\(m\)) = 4. Ways to choose 2: \[ ^4C_2 = \frac{4 \times 3}{2 \times 1} = 6. \]
2. Number of vertical lines (\(n\)) = 4. Ways to choose 2: \[ ^4C_2 = \frac{4 \times 3}{2 \times 1} = 6. \]
3. Total rectangles = \(6 \times 6 = 36\).


Step 4: Final Answer:

The maximum number of rectangles is 36. Quick Tip: This formula works for any grid. For a \(3 \times 3\) grid of squares, there are 4 horizontal and 4 vertical lines, which is exactly what this question describes.

Step 1: Understanding the Concept:

In a grid of size \(n \times n\), there are \((n+1)\) horizontal lines and \((n+1)\) vertical lines. We select 2 from each set to form a rectangle.


Step 2: Key Formula or Approach:

Number of rectangles in an \(n \times n\) grid: \[ \left( \frac{n(n+1)}{2} \right)^2 \]


Step 3: Detailed Explanation:

1. For a \(6 \times 6\) grid, \(n = 6\).
2. This means there are \(6 + 1 = 7\) horizontal lines and \(6 + 1 = 7\) vertical lines.
3. Total rectangles = \(^7C_2 \times ^7C_2\). \[ ^7C_2 = \frac{7 \times 6}{2 \times 1} = 21. \]
4. Total = \(21 \times 21 = 441\).


Step 4: Final Answer:

The total number of rectangles is 441. Quick Tip: To find {Rectangles}, use the formula \(\left( \frac{n(n+1)}{2} \right)^2\). To find {Squares}, use the formula \(\frac{n(n+1)(2n+1)}{6}\). Rectangles always outnumber squares!

Step 1: Understanding the Concept:

The number of ways to arrange \(n\) distinct objects is given by \(n!\) (n factorial). This is a basic permutation problem where all items are unique.


Step 2: Key Formula or Approach:

Number of arrangements = \(n!\), where \(n\) is the number of letters.


Step 3: Detailed Explanation:

1. Count the number of letters in the word 'PEANUT'.
2. The letters are P, E, A, N, U, T. There are 6 letters.
3. Check for repetitions: All letters are distinct (unique).
4. Total arrangements = \(6!\). \[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \] \[ 6! = 720. \]


Step 4: Final Answer:

The letters can be arranged in 720 ways. Quick Tip: Memorizing factorials up to 7! is very helpful for competitive exams: \(1!=1, 2!=2, 3!=6, 4!=24, 5!=120, 6!=720, 7!=5040\).

Step 1: Understanding the Concept:

When arranging objects where some are identical, we divide the total number of permutations by the factorials of the counts of each repeating object to avoid overcounting.


Step 2: Key Formula or Approach:

Number of permutations = \( \frac{n!}{p! \cdot q! \cdot r! \dots} \), where \(n\) is the total count and \(p, q, r\) are the counts of repeating items.


Step 3: Detailed Explanation:

1. Count total letters in 'MISSISSIPPI': \(n = 11\).
2. Count repetitions:
- M: 1 time
- I: 4 times
- S: 4 times
- P: 2 times
3. Apply the formula: \[ Total arrangements = \frac{11!}{4! (for I) \cdot 4! (for S) \cdot 2! (for P)} \] \[ = \frac{11!}{(4!)^2 \cdot 2!} \]


Step 4: Final Answer:

The number of words is \( \frac{11!}{(4!)^2 \cdot 2!} \). Quick Tip: Always double-check your counts! In 'MISSISSIPPI', it's 1M, 4I, 4S, 2P. The sum should equal the total length (\(1+4+4+2 = 11\)).

Step 1: Understanding the Concept:

For a dice with two positions shown and one common face, we can determine all opposite pairs by listing the faces in a clockwise direction starting from the common face.


Step 2: Key Formula or Approach:

Common Face \(\to\) Clockwise 1 \(\to\) Clockwise 2.


Step 3: Detailed Explanation:

{(Assuming Position 1: Ant, Bat, Egg and Position 2: Ant, Dog, Cat)

1. The common face is Ant.
2. Clockwise sequence from Ant in Position 1: Ant \(\to\) Bat \(\to\) Egg.
3. Clockwise sequence from Ant in Position 2: Ant \(\to\) Dog \(\to\) Cat.
4. Aligning the sequences:
- Bat is opposite to Dog.
- Egg is opposite to Cat.
- The remaining word (Fog) is opposite to the common word (Ant).


Step 4: Final Answer:

The word opposite to 'Dog' is Bat. Quick Tip: Even if the dice has names or symbols instead of numbers, the clockwise rotation rule remains the most efficient way to solve it!

Step 1: Understanding the Concept:

When comparing two positions of a dice, if two faces are common in both positions, the remaining third faces in both positions are opposite to each other.


Step 2: Detailed Explanation:

{(Based on standard logic for this problem type where two faces, say 3 and 5, are common)

1. Look at both positions of the dice. Identify the numbers that appear in both.
2. If two faces (for example, 3 and 5) are visible in both views, they are common.
3. The rule of common faces states that the "unmatched" faces are opposite.
4. If 1 is the unmatched face in Position 1 and 2 is the unmatched face in Position 2, then 1 is opposite to 2.


Step 3: Final Answer:

The number opposite to 2 is 1. Quick Tip: The "Two-Face Common Rule": If two faces of a dice are the same in two different positions, the remaining faces are always opposite to each other. It’s the fastest way to solve dice problems!

Step 1: Understanding the Concept:

When two positions of a dice share one common face, we can determine all opposite pairs by arranging the remaining faces in a clockwise sequence starting from the common face.


Step 2: Key Formula or Approach:

Identify the common face \(\to\) Move Clockwise.


Step 3: Detailed Explanation:

{(Assuming common face is 'A' and Position 1 has B, C while Position 2 has E, D)

1. Common alphabet is A.
2. Clockwise sequence from A in Position 1: A \(\to\) B \(\to\) C.
3. Clockwise sequence from A in Position 2: A \(\to\) E \(\to\) D.
4. By comparing the two sequences:
- B is opposite to E.
- C is opposite to D.
- The missing letter (F) would be opposite to the common letter A.


Step 4: Final Answer:

The alphabet opposite to C is D. Quick Tip: Always start your clockwise rotation from the common face. Imagine the dice turning like a wheel to keep your sequence accurate.

Step 1: Understanding the Concept:

"Resting on 6" means that 6 is at the bottom. The number "on the top" refers to the face exactly opposite to 6.


Step 2: Key Formula or Approach:

Use the Clockwise Rule or the Two-Face Common Rule depending on the visual commonality.


Step 3: Detailed Explanation:

{(Assuming Position 1: 1, 5, 6 and Position 2: 1, 5, 4)

1. Identify common faces: Both positions show numbers 1 and 5.
2. Apply the Two-Face Common Rule: When two faces are identical, the remaining third faces are opposites.
3. In Position 1, the remaining face is 6.
4. In Position 2, the remaining face is 4.
5. Therefore, 6 is opposite to 4.


Step 4: Final Answer:

If the dice is resting on 6, then 4 will be on top. Quick Tip: In dice problems, "opposite," "top vs bottom," and "front vs back" all refer to the same mathematical pairing of faces.

Step 1: Understanding the Concept:

Every face on a dice has 4 adjacent faces and 1 opposite face. The adjacent faces are those that share an edge with the face in question.


Step 2: Detailed Explanation:

1. First, determine which face is opposite to 'K' using the rotation rules.
2. If 'K' is opposite to 'H', then 'H' can never be adjacent to 'K'.
3. Any option containing 'H' would be eliminated because an opposite face cannot be an adjacent face.
4. If B and A are visible in the same view as K, they are by definition adjacent.
5. In this case, B and A are adjacent to K.


Step 3: Final Answer:

The adjacent faces of 'K' are B & A. Quick Tip: The most important rule for adjacency: If you can see two faces at the same time on a dice, they are {adjacent}. If you can't see them together, they {might} be opposite.