Arrangement question?

25.
1-digit even numbers: 4 and 6
No. of 1-digit even numbers = 2

2-digit even numbers:
The units digit is "4" or "6" (₂P₁). Assign 1 of the rest 4 digits to the tens digit place (₄P₁).
No. of 2-digit even numbers = ₂P₁ × ₄P₁ = 2 × 4 = 8

3-digit even numbers:
The units digit is "4" or "6" (₂P₁). Assign 2 of the rest 4 digits to the rest 2 digit places (₄P₂).
No. of 3-digit even numbers = ₂P₁ × ₄P₂ = 2 × 12 = 24

4-digit even numbers:
The units digit is "4" or "6" (₂P₁). Assign 3 of the rest 4 digits to the rest 3 digit places (₄P₃).
No. of 3-digit even numbers = ₂P₁ × ₄P₃ = 2 × 24 = 48

5-digit even numbers:
The units digit is "4" or "6" (₂P₁). Assign the rest 4 digits to the rest 4 digit places (₄P₄).
No. of 3-digit even numbers = ₂P₁ × ₄P₄ = 2 × 24 = 48

Total no. of even numbers formed
= 2 + 8 + 24 + 48 + 48
= 130

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26.
(a)
There are 1 red peg, 1 white peg, 1 black peg, 1 green peg, 1 blue peg and 1 yellow peg.
Out of the 6 pegs, place 4 pegs in the 4 holes (₆P₄).

Number of arrangements = ₆P₄ = 360

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(b)
Placing 4 pegs in the 4 holes is ₄P₄. Divided by 2!2! because there are 2 red pegs and 2 white pegs.

No. of arrangements = ₄P₄/(2!2!) = 24/4 = 6

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(c)
Placing 4 pegs into the 4 holes is ₄P₄. Divided by 2! because there are 2 red pegs.

No. of arrangements = ₄P₄/(2!2!) = 24/4 = 6

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(d)
There are 2 red pegss, 2 white pegs, 2 black pegs, 2 green pegs, 2 blue pegs and 2 yellow pegs.

4 pegs with 2 colours:
Firstly, choose the 2 colours from the 6 colours(₆C₂). Placing 4 pegs into the 4 holes is ₄P₄. Divided by 2!2! because each pair of pegs are painted with the same colour.
No of arrangements for 4 pegs with 2 colours
= ₆C₂ × ₄P₄ / (2!2!)
= 15 × 24 / 4
= 90

4 pegs with 3 colours:
Firstly, choose 1 colour from the 6 colours for 2 pegs (₆C₁). Then, choose other 2 colours from the rest 5 colours for other 2 pegs (₅C₂). Placing 4 pegs into the 4 holes is ₄P₄. Divided by 2! because one pair of pegs are painted with the same colour.
No of arrangements for 4 pegs with 3 colours
= ₆C₁ × ₅C₂ × ₄P₄ / 2
= 6 × 10 × 24 / 2
= 720

4 pegs with 4 colours:
Choose 4 colours from the 6 colours for 4 pegs (₆C₄). Placing 4 pegs into the 4 holes is ₄P₄
No of arrangements for 4 pegs with 4 colours
= ₆C₄ × ₄P₄
= 15 × 24
= 360

Total no. of arrangements
= 90 + 720 + 360
= 1170

4 and 6 are the only even numbers
and
the final digit of an even number is always an even number.

SO: you have
single digit - 2 even numbers
double digit - 4 x 2 even numbers
etc.