Time: Mon 9/11/17 3 pm

Suppose we have a sequence of quantities that we want to multiply together, but can only multiply two at a time. We can express the various ways of doing so by counting the number of different ways to parenthesize the sequence. For example, here are the possibilities for products of 1, 2, 3 and 4 elements:

- 1 element of 1 possibility

`a`

- 2 elements of 1 possibility

`ab`

- 3 elements of 2 possibilities

`(ab)c`

`a(bc)`

- 4 elements of 5 possibilities

`a(b(cd))`

`a((bc)d)`

`(ab)(cd)`

`(a(bc))d`

`((ab)c)d`

Assume, as in the table above, that we don’t want to reorder elements.

Define a function `count_groupings`

that takes a positive integer `n`

and returns the number of ways of parenthesizing the product of `n`

numbers. (You might not need to use all lines.)

```
def count_groupings(n):
"""
For N >= 1, the number of distinct parenthesizations of a product of N items.
>>> count_groupings(1)
1
>>> count_groupings(2)
1
>>> count_groupings(3)
2
>>> count_groupings(4)
5
>>> count_groupings(5)
14
"""
if n == 1:
return
i =
while :
i += 1
return
```

Draw the the environment diagram that results from executing the code below until the entire program is finished or an error occurs.

```
y = 3
def out(h, m):
y = 5 * m
def inner():
return y
if m == 0:
return h
else:
return out(inner, m - 1)
v = out(None, 1)()
```

**IMPORTANT DEFINITION** Each digit in a non-negative integer `n`

has a digit position. Digit positions begin at `0`

and count from the right-most digit of `n`

. For example, in `568789`

, the digit `9`

is at position `0`

and digit `7`

is at position `2`

. The digit `8`

appears at both positions `1`

and `3`

.

Implement the find_digit function, which takes a non-negative integer

`n`

and a digit`d`

greater than`0`

and less than`10`

. It returns the largest (left-most) position in`n`

at which digit`d`

appears. If`d`

does not appear in`n`

, then`find_digit`

returns`False`

. You may not use recursive calls.`def find_digit(n, d): """ Return the largest digit position in n for which d is the digit. >>> find_digit(567, 7) 0 >>> find_digit(567, 5) 2 >>> find_digit(567, 9) False >>> find_digit(568789, 8) # 8 appears at positions 1 and 3 3 """ i, k = 0, while n: n, last = n // 10, n % 10 if last == : i = i + 1 return`

Find all values of

`y`

between 1 and 9 (inclusive) for which the final expression below evaluates to`True`

. Assume that`find_digit`

is implemented correctly.`def compose1(f, g): def h(x): return f(g(x)) return h f = lambda x: find_digit(234567, x) compose1(f, f)(y) == y`

Add parentheses and single-digit integers in the blanks below so that the expression on the second line evaluates to `2015`

. You may only add parentheses and single-digit integers. You may leave some blanks empty.

```
lamb = lambda lamb: lambda: lamb + lamb
lamb(1000)__________ + (lambda b, c: b__________ * b__________ - c__________)(lamb(__________), 1)__________
```